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This article will explain the domain and range of a function mean and how to calculate the two quantities. Before getting into the topic of domain and range, let's briefly describe what a function is.

In mathematics, we can compare a function to a machine that generates some output in correlation to a given input. By taking an example of a coin stamping machine, we can illustrate a function's meaning as follows.

When you insert a coin into the coin stamping machine, the result is a stamped and flattened piece of metal. By considering a function, we can relate the coin and the flattened piece of metal with the domain and range. In this case, a function is considered to be the coin stamping machine.

Just like the coin stamping machine, which can only produce a single flattened piece of metal at a time, a function works in the same manner by giving out one result at a time.

History of a function

The idea of a function was introduced in the early seventeenth century when Rene Descartes (1596-1650) used the concept in his book Geometry (1637) to model mathematical problems.

Fifty years later, after the publication of Geometry, Gottfried Wilhelm Leibniz (1646-1716) introduced the term "function." Later, Leonhard Euler (1707-1783) played a big role by introducing the technique of function notion, y = f (x).

Real-life application of a function

Functions are very useful in mathematics because they allow us to model real-life problems into a mathematical format.

Here are a few examples of the application of a function.

The circumference of a circle is a function of its diameter or radius. We can mathematically represent this statement as:

C(d) =dπ or C(r)=2π⋅r

The length of the shadow of an object is a function of its height.

The location of a moving object such as a car is a function of time.

The temperature of a body is based on several factors and inputs.

The compound or simple interest is a function of the time, principal, and interest rate.

The height of an object is a function of his/her age and body weight.

Having learned about a function now can proceed to how to calculate the domain and the range of a function.

What is the Domain and Range of a Function?

The domain of a function is the input numbers that, when plugged into a function, the result is defined. In simple words, we can define the domain of a function as the possible values of x that will make an equation true.

Some of the instances that will not make a valid function are when an equation is being divided by zero or a negative square root.

For example, f( x) = x ² is a valid function because, no matter what value of x can be substituted into an equation, there is always a valid answer. For this reason, we can conclude that the domain of any function is all real numbers.

The range of a function is defined as a set of solutions to the equation for a given input. In other words, the range is the output or y value of a function. There is only one range for a given function.

How to use interval notations to specify Domain and Range?

Since the range and domain of a function are usually expressed in interval notation, it's important to discuss the concept of interval notation.
The procedure for doing interval notation include:

Write the numbers separated by a comma in ascending order.

Enclose the numbers using parentheses () to show that an endpoint value is not included.

Use brackets [] to enclose the numbers when the endpoint value is included.

How to Find the Domain and Range of a Function?

We can determine the domain of a function either algebraically or by the graphical method. To calculate the domain of a function algebraically, you solve the equation to determine the values of x.

Different types of functions have their own methods of determining their domain.
Let's examine these types of functions and how to calculate their domain.
How to find the domain for a function with no denominator or radicals?

Let's see a few examples below to understand this scenario.
Example 1
Find the domain of f (x) = 5x − 3

The domain of a linear function is all real numbers, therefore,

Domain: (−∞, ∞)

Range: (−∞, ∞)
A function with a radical Example 2
Find the domain of the function f(x)=−2x 2 + 12x + 5

The function f(x) = −2x 2 + 12x + 5 is a quadratic polynomial, therefore, the domain is (−∞, ∞)

How to find the domain for a rational function with a variable in the denominator?

To find the domain of this type of function, set the denominator to zero and calculate the variable's value.

Let's see a few examples below to understand this scenario.
Example 3
Determine the domain of x−4/ (x 2 −2x−15)

Set the denominator to zero and solve for x

⟹ x 2 − 2x - 15 = (x − 5) (x + 3) = 0

Hence, x = −3, x = 5

For the denominator not to be zero, we need to avoid the numbers −3 and 5. Therefore, the domain is all real numbers except −3 and 5.
Example 4
Calculate the domain and the range of the function f(x) = -2/x.

Set the denominator to zero.

⟹ x = 0

Therefore, domain: All real numbers except 0.

The range is all real values of x except 0.
Example 5
Find the domain and range of the following function.

f(x) = 2/ (x + 1)

Set the denominator equal to zero and solve for x.

x + 1 = 0

= -1

Since the function is undefined when x = -1, the domain is all real numbers except -1. Similarly, the range is all real numbers except 0

How to the domain for a function with a variable inside a radical sign?

To find the domain of the function, the terms inside the radical are set the inequality of > 0 or ≥ 0. Then, the value of the variable is determined.

Let's see a few examples below to understand this scenario.
Example 6
Find the domain of f(x) = √ (6 + x - x ²)

To avoid the square roots of negative numbers, we set the expression inside the radical sign to ≥ 0.

⟹ x 2 - x - 6= (x - 3) (x +2) = 0

Therefore, the function is zero if x = 3 or x = -2

Hence the domain: [−2, 3]
Example 7
Find the domain of f(x) =x/√ (x 2 - 9)

Set the expression within the radical sign to x 2 - 9 > 0
Solve for the variable to get;

x = 3 or - 3

Therefore, Domain: (−∞, −3) & (3, ∞)
Example 8
Find the domain of f(x) = 1/√ (x ² -4)

By factoring the denominator, we get x ≠ (2, - 2).

Test your answer by plugging -3 into the expression within the radical sign.

also try with zero

⟹ 0 ² - 4 = -4, therefore number between 2 and -2 are invalid

Try number above 2

⟹ 3 ² - 4 = 5. This one is valid.

Hence, the domain = (-∞, -2) U (2, ∞)

How to find the domain of a function using the natural logarithm (ln)?

To find the domain of a function using natural log, set the terms within the parentheses to >0 and then solve.

Let's see an example below to understand this scenario.
Example 9
Find the domain of the function f(x) = ln (x - 8)

⟹ x - 8 > 0

⟹ x - 8 + 8 > 0 + 8

⟹ x > 8

Domain:(8, ∞)

How to find the domain and range of a relation?

A relation is an asset of x and y coordinates. To find the domain and range in a relation, just list the x and y values, respectively.

Let's see a few examples below to understand this scenario.
Example 10
State the domain and range of the relation {(2, -3), (4, 6), (3, -1), (6, 6), (2, 3)}

List the x values. Domain: {2, 3, 4, 6}

List the y values. range: {-3, -1, 3, 6}
Example 11
Find the domain and range of the relation {(-3, 5), (-2, 5), (-1, 5), (0, 5), (1, 5), (2, 5)}

The domain is {-3, -2, -1, 0, 1, 2} and the range is {5}
Example 12
Given that R = {(4, 2) (4, -2), (9, 3) (9, -3)}, find the domain and range of R.

The domain is a list of first values, therefore, D= {4, 9} and the range = {2, -2, 3, -3}

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Domain and Range of a Function - Explanation & Examples

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This article will explain the domain and range of a function mean and how to calculate the two quantities. Before getting into the topic of domain and range, let's briefly describe what a function is. In mathematics, we can compare a function to a machine that generates some output in correlation to a given input.

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Biography

René Descartes was a philosopher whose work, La Géométrie , includes his application of algebra to geometry from which we now have Cartesian geometry.

René Descartes' parents were Joachim Descartes (1563-1640) and Jeanne Brochard (1566-1597). Joachim, the son of the medical doctor Pierre Descartes (1515-1566), studied law and was a counsellor in the Parliament of Brittany which sat at Rennes. Jeanne was the daughter of the military man René Brochard who formed part of the garrison stationed at Poitiers. One of Jeanne's brothers, also named René Brochard, became one of René Descartes' two godfathers; René Descartes was named after his godfather René Brochard. Jeanne's widowed mother, Jeanne Sain Brochard, lived at La Haye, near Tours, and it was in her home that René was born. Joachim and Jeanne Descartes were married on 15 January 1589 and lived at Châtellerault. They had two surviving children older than René, a girl named Jeanne (born 1590) and a boy named Pierre (born 1591). René was baptised in the Roman Catholic Church of Saint George in La Haye when he was four days old. His mother died in childbirth a year after he was born and the boy, born at the time of her death, also died. At this time, René was sent back to his grandmother's home in La Haye where he was cared for by Jeanne Sain Brochard. Joachim Descartes remarried in 1600 to Anne Morin and they had a boy named Joachim (born 1602) and a girl named Anne (born 1611). René therefore had a older brother and an older sister, as well as a younger half-brother and younger half-sister. He did not return to live with his father and step-mother, however, but continued living with his grandmother at La Haye. Now René's health was poor when he was a child. Throughout his childhood, up to his twenties, he was pale and had a persistent cough which was probably due to tuberculosis. It seems likely that he inherited these health problems from his mother.

Descartes was educated at the Jesuit college of La Flèche in Anjou. He entered the college at Easter 1607 at the age of eleven years where he became a boarder. [There is some uncertainty here, with some biographers claiming that he entered the College two years earlier.] The college had opened in January 1604 so was a comparatively new school. He studied there taking courses in classics, logic and traditional Aristotelian philosophy. He also learnt mathematics from the books of Clavius, while studying all the branches of mathematics, namely arithmetic, geometry, astronomy and music. While in the school his health was poor and, instead of rising at 5 a.m. like the other boys, he was granted permission to remain in bed until 11 o'clock in the morning, a custom he maintained until the year of his death. In his final years at the school he studied natural philosophy, metaphysics and ethics. He left the La Flèche College in 1614. School had made Descartes understand how little he knew, the only subject which was satisfactory in his eyes was mathematics. This idea became the foundation for his way of thinking, and was to form the basis for all his works.

See Descartes' own words describing his schooling at THIS LINK.

Comparatively little is known of Descartes' life between 1614 and 1618. He spent a while in Paris, apparently keeping very much to himself, and some have speculated that he might have suffered some sort of a breakdown at this time. Then he studied at the University of Poitiers, receiving a law degree from Poitiers in 1616. He took the law degree to comply with his father's wishes but he quickly decided that this was not the path he wanted to follow. He wrote in Discourse on the Method:-

I entirely abandoned the study of letters, resolving to seek no knowledge other than that which could be found in myself or else in the great book of the world. I spent the rest of my youth travelling, visiting courts and armies, mixing with people of diverse temperaments and ranks, gathering various experiences, testing myself in the situations which fortune offered me, and at all times reflecting on whatever came my way so as to derive some profit from it.

He may have returned to Paris before he enlisted in the military school at Breda in 1618, becoming a volunteer in the army of Maurice of Nassau. While in Breda his formal study was of military engineering but he started studying mathematics and mechanics under the Dutch scientist Isaac Beeckman, and began to seek a unified science of nature. Advised by Beeckman, he began considering mechanical problems. While in Holland, he wrote to Beeckman in March 1619 about his new ideas:-

[I want to promote a] completely new science by which all questions in general may be solved that can be proposed about any kind of quantity, continuous as well as discrete. But each according to its own nature. ... In arithmetic, for instance, some questions can be solved by rational numbers, some by surd numbers, and others can be imagined but not solved. For continuous quantity I hope to prove that, similarly, certain problems can be solved by using only straight or circular lines, that some problems require other curves for their solution, but still curves which arise from one single motion and which therefore can be traced by the new compasses, which I consider to be no less certain and geometrical than the usual compasses by which circles are traced; and, finally, that other problems can be solved by curved lines generated by separate motions not subordinate to one another.

After this time in Holland he left the service of Maurice of Nassau and travelled through Europe with the plan to join the army of Maximilian of Bavaria. In 1619 he joined the Bavarian army and was stationed in Ulm. An important event in his life was three dreams he had in November 1619. These he believed were sent by a divine spirit with the intention of revealing to him a new approach to philosophy. The ideas from these dreams would dominate much of his work from that time on.

For Descartes' own description of the ideas that he developed at this time see THIS LINK.

While serving Maximilian of Bavaria, Descartes was present as an official observer with the Catholic League at the Battle of the White Mountain near Prague in November 1620. After this he left the army but since the plague was ravaging in Paris he could not return there but instead began a period of travel.

From 1620 to 1628 Descartes travelled through Europe, spending time in Bohemia (1620), Hungary (1621), Germany, Holland and France (1622-23). He spent time in 1623 in Paris where he made contact with Marin Mersenne, an important contact which kept him in touch with the scientific world for many years, and with Claude Mydorge. From Paris he travelled through Switzerland to Italy where he spent some time in Venice and in Rome, then he returned to France again (1625). He renewed his acquaintance with Mersenne and Mydorge, and met Girard Desargues. His Paris home became a meeting place for philosophers and mathematicians and steadily became more and more busy. By 1628 Descartes, tired of the bustle of Paris, the house full of people, and of the life of travelling he had before, decided to settle down where he could work in solitude. He gave much thought to choosing a country suited to his nature and he chose Holland. What he longed for was somewhere peaceful where he could work away from the distractions of a city such as Paris yet still have access to the facilities of a city. It was a good decision which he did not seem to regret over the next twenty years. He told Mersenne where he was living so that he might keep in touch with the mathematical world, but otherwise he kept his place of residence a secret.

Soon after he settled in Holland, Descartes began work on his first major treatise on physics, Le Monde, ou Traité de la Lumière . He wrote to Mersenne in October 1629:-

[The foundations of physics] is the topic which I have studied more than any other and in which, thank God, I have not altogether wasted my time. At least I think that I have found how to prove metaphysical truths in a manner which is more evident than the proofs of geometry - in my opinion, that is: I do not know if I shall be able to convince others of it. During my first nine months in this country I worked on nothing else.

This work was near completion when news that Galileo was condemned to house arrest reached him. He, perhaps wisely, decided not to risk publication and the work was published, only in part, after his death. He explained later his change of direction saying:-

... in order to express my judgement more freely, without being called upon to assent to, or to refute the opinions of the learned, I resolved to leave all this world to them and to speak solely of what would happen in a new world, if God were now to create ... and allow her to act in accordance with the laws He had established.

For Descartes' own description of the contents of Le Monde, ou Traité de la Lumière see THIS LINK.

In Holland, Descartes had a number of scientific friends as well as continued contact with Mersenne. His friendship with Beeckman continued and he also had contact with Mydorge, Hortensius, Huygens and Frans van Schooten (the elder). Langer describes Descartes' life in Holland:-

As throughout his life he continued to do his work abed in the mornings. His evenings he generally devoted to the consideration of his correspondence, which was mainly scientific, rarely personal, and of which he was painstakingly careful, while the intermediate part of the day he gave to relaxation. In matters of money he was neither extravagant nor parsimonious, showing himself in this respect a true philosopher. He always did some entertaining, now more, now less, professing to find considerable enjoyment in conversation, though he was himself rather taciturn.

Descartes was pressed by his friends to publish his ideas and, although he was adamant in not publishing Le Monde , he wrote a treatise on science under the title Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences . Three appendices to this work were La Dioptrique , Les Météores , and La Géométrie . The treatise was published at Leiden in 1637 and Descartes wrote to Mersenne saying:-

I have tried in my "Dioptrique" and my "Météores" to show that my Méthode is better than the vulgar, and in my "Géométrie" to have demonstrated it.

The work describes what Descartes considers is a more satisfactory means of acquiring knowledge than that presented by Aristotle's logic. Only mathematics, Descartes feels, is certain, so all must be based on mathematics.

La Dioptrique is a work on optics and, although Descartes does not cite previous scientists for the ideas he puts forward, in fact there is little new. However his approach through experiment was an important contribution.

Les Météores is a work on meteorology and is important in being the first work which attempts to put the study of weather on a scientific basis. However many of Descartes' claims are not only wrong but could have easily been seen to be wrong if he had done some easy experiments. For example Roger Bacon had demonstrated the error in the commonly held belief that water which has been boiled freezes more quickly. However Descartes claims:-

... and we see by experience that water which has been kept on a fire for some time freezes more quickly than otherwise, the reason being that those of its parts which can be most easily folded and bent are driven off during the heating, leaving only those which are rigid.

Despite its many faults, the subject of meteorology was set on course after publication of Les Météores particularly through the work of Boyle, Hooke and Halley.

La Géométrie is by far the most important part of this work. In Scott summarises the importance of this work in four points:-

1. He makes the first step towards a theory of invariants, which at later stages derelativises the system of reference and removes arbitrariness.

2. Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all.

3. Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method.

4. Not only can questions of solvability and geometrical possibility be decided elegantly, quickly and fully from the parallel algebra, without it they cannot be decided at all.

Some extracts from Descartes' La Géométrie are at THIS LINK.
Some ideas in La Géométrie may have come from earlier work of Oresme but in Oresme's work there is no evidence of linking algebra and geometry. Wallis in Algebra (1685) strongly argues that the ideas of La Géométrie were copied from Harriot. Wallis writes:-

... the Praxis was read by Descartes, and every line of Descartes' analysis bears token of the impression.

There seems little to justify Wallis's claim, which was probably made partly through patriotism but also through his just desires to give Harriot more credit for his work. Harriot's work on equations, however, may indeed have influenced Descartes who always claimed, clearly falsely, that nothing in his work was influenced by the work of others.

Descartes' Meditations on First Philosophy, was published in 1641, designed for the philosopher and for the theologian. It consists of six meditations, Of the Things that we may doubt, Of the Nature of the Human Mind, Of God: that He exists, Of Truth and Error, Of the Essence of Material Things, Of the Existence of Material Things and of the Real Distinction between the Mind and the Body of Man. However many scientists were opposed to Descartes' ideas including Arnauld, Hobbes and Gassendi.

The most comprehensive of Descartes' works, Principia Philosophiae was published in Amsterdam in 1644. In four parts, The Principles of Human Knowledge, The Principles of Material Things, Of the Visible World and The Earth, it attempts to put the whole universe on a mathematical foundation reducing the study to one of mechanics. This is an important point of view and was to point the way forward. Descartes did not believe in action at a distance. Therefore, given this, there could be no vacuum around the Earth otherwise there was no way that forces could be transferred. In many ways Descartes' theory, where forces work through contact, is more satisfactory than the mysterious effect of gravity acting at a distance. However Descartes' mechanics leaves much to be desired. He assumes that the universe is filled with matter which, due to some initial motion, has settled down into a system of vortices which carry the sun, the stars, the planets and comets in their paths. Despite the problems with the vortex theory it was championed in France for nearly one hundred years even after Newton showed it was impossible as a dynamical system. As Brewster, one of Newton's 19 th century biographers, puts it:-

Thus entrenched as the Cartesian system was ... it was not to be wondered at that the pure and sublime doctrines of the 'Principia' were distrustfully received ... The uninstructed mind could not readily admit the idea that the great masses of the planets were suspended in empty space, and retained their orbits by an invisible influence ...

Pleasing as Descartes' theory was, even the supporters of his natural philosophy such as the Cambridge metaphysical theologian Henry More, found objections. Certainly More admired Descartes, writing:-

I should look upon Des-Cartes as a man most truly inspired in the knowledge of Nature, than any that have professed themselves so these sixteen hundred years...

However between 1648 and 1649 they exchanged a number of letters in which More made some telling objections. Descartes however in his replies makes no concessions to More's points. More went on to ask:-

Why are not your vortices in the form of columns or cylinders rather than ellipses, since any point of the axis of a vortex is as it were a centre from which the celestial matter recedes with, as far as I can see, a wholly constant impetus? ... Who causes all the planets not to revolve in one plane (the plane of the ecliptic)? ... And the Moon itself, neither in the plane of the Earth's equator nor in a plane parallel to this?

In 1644, the year his Meditations were published, Descartes visited France. He returned again in 1647, when he met Pascal and argued with him that a vacuum could not exist, and then again in 1648.

In 1649 Queen Christina of Sweden persuaded Descartes to go to Stockholm. However the Queen wanted to draw tangents at 5 a.m. and Descartes broke the habit of his lifetime of getting up at 11 o'clock. After only a few months in the cold northern climate, walking to the palace for 5 o'clock every morning, he died of pneumonia.

After his death an unfinished manuscript entitled Regulae ad directionem ingenii was found in his papers. Only the first 21 of the Rules were presented, the last three being only given by their intended titles. Sadly, the original manuscript has been lost and only copies remain. Here is a short extract from the manuscript:-

I would not value these Rules so highly if they were good only for solving those pointless problems with which arithmeticians and geometers are inclined to while away their time, for in that case all I could credit myself with achieving would be to dabble in trifles with greater subtlety than they. I shall have much to say below about figures and numbers, for no other disciplines can yield illustrations as evident and certain as these. But if one attends closely to my meaning, one will readily see that ordinary mathematics is far from my mind here, that it is quite another discipline I am expounding, and that these illustrations are more its outer garments than its inner parts. This discipline should contain the primary rudiments of human reason and extend to the discovery of truths in any field whatever. Frankly speaking, I am convinced that it is a more powerful instrument of knowledge than any other with which human beings are endowed, as it is the source of all the rest.

We should end this biography by saying a little more about Descartes as a person. In Langer describes Descartes' appearance and personality:-

In appearance Descartes was a small man of rather slight figure with a large head. His nose was prominent, his lower lip somewhat protruding, his beard and moustache of a semi-military type, and his hair growing down upon his forehead almost to his eyebrows. He wore a wig of natural colour to which he always gave fastidious attention, as he did also to his clothes which were now invariably of black cloth. In demeanour he was generally cheerful, rarely gay. His manners were always refined, gentle, and polite, and his temper tranquil and easy. As a personality he was proud, somewhat aristocratically reserved, sensitive, a bit angular, and, though a shade domineering, was pre-eminently obliging.

He always was well dressed, and wore a sword. He was not industrious; he worked short hours, and read little. When he went to Holland he took few books with him, but among them were the Bible and Thomas Aquinas. His work seems to have been done with great concentration during short periods; but perhaps, to keep up the appearance of a gentlemanly amateur, he may have pretended to work less than in fact he did, for otherwise his achievements seem scarcely credible.

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René Descartes was a philosopher whose work, La Géométrie , includes his application of algebra to geometry from which we now have Cartesian geometry. René Descartes' parents were Joachim Descartes (1563-1640) and Jeanne Brochard (1566-1597). Joachim, the son of the medical doctor Pierre Descartes (1515-1566), studied law and was a counsellor in the Parliament of Brittany which sat at Rennes.

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How to Use the Google Translate Mobile App

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Cómo usar Google Translate en tu celular

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Domain and Range

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In this activity, students describe the domain and range of six relationships. (First verbally and later algebraically.) Later, students use movable points to create three functions whose domain and range match specific criteria. French translation courtesy of Litissia Abi Abdallah et Joce: https://teacher.desmos.com/activitybuilder/custom/591f122dfbd2350b804d2345

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1. Function/Función
2. Domain/El Dominio
3. Range/ El Rango
4. Inequality Notation/Notación de Desigualdad

5. Interval Notation/Notación de Intervalos

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1. Students will be able to identify functions.
2. Students will be able to explain what domain and range are.
3. Students will be able to solve for the domain and range of a function when given an equation, word problem or a graph.

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1. Students will be able to summarize solving for domain and range, using both English and their native language.

2. Students will be able to create a function that applies to something in their life.

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Capítulo 2 - Funciones (Página 96, o 117)

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1. Students will be able to read descriptions of what makes a function a function.
2. Students will be able to describe the relationship between domain and range in English.
3.  Students will be able to explain their steps of how they found the domain and range of a function given an equation or a graph in the language they feel most comfortable with.

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Link to Insta video: https://www.instagram.com/reel/Cere9J9lL9-/?utm_source=ig_web_copy_link&igshid=MzRlODBiNWFlZA==

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Link to Insta: https://www.instagram.com/reel/CewhaQ-lpJJ/?utm_source=ig_web_copy_link&igshid=MzRlODBiNWFlZA==

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Quizizz - The world's most engaging learning platform

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Find and create gamified quizzes, lessons, presentations, and flashcards for students, employees, and everyone else. Get started for free!

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Jamboard: https://jamboard.google.com/d/1douQyIT4IVIuyHXemPPvmNWgMQrd3Ab5kxB_kQfJ_l8/edit?usp=sharing

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Images/GIPHS representing the vocabulary and/or the Unit topic.  

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Images/GIPHS representing the vocabulary and/or the Unit topic.  

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1. Students will be able to know how to use Google Translate to help them translate definitions, word problems, etc.
2. Students will be able to know how to turn on the captions on a Youtube video if needed.

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Domain and Range

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Dominio y Rango de una función #aprendeentiktok #transformar2022 #matematicas #edutok #matematicastiktok #math #matematica #algebra #calculo #calculodiferencial #funciones #relaciones #conjuntos #dominioyrango

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@willymath93
Dominio y Rango de una función #aprendeentiktok #transformar2022 #matematicas #edutok #matematicastiktok #math #matematica #algebra #calculo #calculodiferencial #funciones #relaciones #conjuntos #dominioyrango
♬ sonido original - WillyMath

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Create a new document and edit with others at the same time -- from your computer, phone or tablet. Get stuff done with or without an internet connection. Use Docs to edit Word files. Free from Google.

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Functions or not a Function/Función o no Función

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Functions or not a Function/Función o no Función

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Functions

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Function or nah? #function #mathfunctions #algebrs #functions #mathhelper #askme #mathmethics #mathtutors #fyp #prealgebra

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@homemademathematics
Function or nah? #function #mathfunctions #algebrs #functions #mathhelper #askme #mathmethics #mathtutors #fyp #prealgebra
♬ Skate - Trees and Lucy

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Función

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Responder a @el_vinke19 Mañana subiré como identificar si una gráfica es una función o no! 😃 #matematicas #funcion #calculo #examen

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@matematicascorps
Responder a @el_vinke19 Mañana subiré como identificar si una gráfica es una función o no! 😃 #matematicas #funcion #calculo #examen
♬ sonido original - Matematicas Corps

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#aprendeentiktok #fypシ #fyp #matematicastiktok #math #edutok #viral #matematicas #matematica #edutuber #transformar2022 #algebra #calculo #calculodiferencial #funciones 

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@willymath93
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Interval Notation #algebra1 #algebrahelp #inequalities #intervalnotation #thatsit 

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@thatsitalgebra
Interval Notation #algebra1 #algebrahelp #inequalities #intervalnotation #thatsit
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En el siguiente problema, escribe la desigualdad utilizando la notación de intervalo e ilústrala. 4≤𝑥≤6

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@alberto_argon
En el siguiente problema, escribe la desigualdad utilizando la notación de intervalo e ilústrala. 4≤𝑥≤6
♬ sonido original - José Alberto Arrocena González

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Desmos: Finding Domain and Range - French

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Desmos: Finding Domain and Range - French

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Dans cette activité, les élèves doivent décrire le domaine et l'image dans six relations. Par la suite, les élèves utilisent des points mobiles pour créer trois fonctions avec des critères spécifiques pour le domaine et l'image. Traduction: Litissia Abi Abdallah et Jocelyn Dagenais

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Identifying input and output

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Algebra 1   Real World Examples of Functions

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Real World Applications - Spanish Version

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Real World Applications - Spanish Version

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Mire solo desde el principio hasta los cuatro minutos y 28 segundos.

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Real World Applications and Interpreting Graphs (Spanish version)

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Quick Info

Born

1 July 1646
Leipzig, Saxony (now Germany)

Died

14 November 1716
Hannover, Hanover (now Germany)

Summary

Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

Biography

Gottfried Leibniz was the son of Friedrich Leibniz, a professor of moral philosophy at Leipzig. Friedrich Leibniz :-

...was evidently a competent though not original scholar, who devoted his time to his offices and to his family as a pious, Christian father.
Leibniz's mother was Catharina Schmuck, the daughter of a lawyer and Friedrich Leibniz's third wife. However, Friedrich Leibniz died when Leibniz was only six years old and he was brought up by his mother. Certainly Leibniz learnt his moral and religious values from her which would play an important role in his life and philosophy.

At the age of seven, Leibniz entered the Nicolai School in Leipzig. Although he was taught Latin at school, Leibniz had taught himself far more advanced Latin and some Greek by the age of 12. He seems to have been motivated by wanting to read his father's books. As he progressed through school he was taught Aristotle's logic and theory of categorising knowledge. Leibniz was clearly not satisfied with Aristotle's system and began to develop his own ideas on how to improve on it. In later life Leibniz recalled that at this time he was trying to find orderings on logical truths which, although he did not know it at the time, were the ideas behind rigorous mathematical proofs. As well as his school work, Leibniz studied his father's books. In particular he read metaphysics books and theology books from both Catholic and Protestant writers.

In 1661, at the age of fourteen, Leibniz entered the University of Leipzig. It may sound today as if this were a truly exceptionally early age for anyone to enter university, but it is fair to say that by the standards of the time he was quite young but there would be others of a similar age. He studied philosophy, which was well taught at the University of Leipzig, and mathematics which was very poorly taught. Among the other topics which were included in this two year general degree course were rhetoric, Latin, Greek and Hebrew. He graduated with a bachelors degree in 1663 with a thesis De Principio Individui which:-
... emphasised the existential value of the individual, who is not to be explained either by matter alone or by form alone but rather by his whole being.
In this there is the beginning of his notion of "monad". Leibniz then went to Jena to spend the summer term of 1663.

At Jena the professor of mathematics was Erhard Weigel but Weigel was also a philosopher and through him Leibniz began to understand the importance of the method of mathematical proof for subjects such as logic and philosophy. Weigel believed that number was the fundamental concept of the universe and his ideas were to have considerable influence of Leibniz. By October 1663 Leibniz was back in Leipzig starting his studies towards a doctorate in law. He was awarded his Master's Degree in philosophy for a dissertation which combined aspects of philosophy and law studying relations in these subjects with mathematical ideas that he had learnt from Weigel. A few days after Leibniz presented his dissertation, his mother died.

After being awarded a bachelor's degree in law, Leibniz worked on his habilitation in philosophy. His work was to be published in 1666 as Dissertatio de arte combinatoria . In this work Leibniz aimed to reduce all reasoning and discovery to a combination of basic elements such as numbers, letters, sounds and colours.

Despite his growing reputation and acknowledged scholarship, Leibniz was refused the doctorate in law at Leipzig. It is a little unclear why this happened. It is likely that, as one of the younger candidates and there only being twelve law tutorships available, he would be expected to wait another year. However, there is also a story that the Dean's wife persuaded the Dean to argue against Leibniz, for some unexplained reason. Leibniz was not prepared to accept any delay and he went immediately to the University of Altdorf where he received a doctorate in law in February 1667 for his dissertation De Casibus Perplexis .

Leibniz declined the promise of a chair at Altdorf because he had very different things in view. He served as secretary to the Nuremberg alchemical society for a while (see ) then he met Baron Johann Christian von Boineburg. By November 1667 Leibniz was living in Frankfurt, employed by Boineburg. During the next few years Leibniz undertook a variety of different projects, scientific, literary and political. He also continued his law career taking up residence at the courts of Mainz before 1670. One of his tasks there, undertaken for the Elector of Mainz, was to improve the Roman civil law code for Mainz but :-
Leibniz was also occupied by turns as Boineburg's secretary, assistant, librarian, lawyer and advisor, while at the same time a personal friend of the Baron and his family.
Boineburg was a Catholic while Leibniz was a Lutheran but Leibniz had as one of his lifelong aims the reunification of the Christian Churches and :-
... with Boineburg's encouragement, he drafted a number of monographs on religious topics, mostly to do with points at issue between the churches...
Another of Leibniz's lifelong aims was to collate all human knowledge. Certainly he saw his work on Roman civil law as part of this scheme and as another part of this scheme, Leibniz tried to bring the work of the learned societies together to coordinate research. Leibniz began to study motion, and although he had in mind the problem of explaining the results of Wren and Huygens on elastic collisions, he began with abstract ideas of motion. In 1671 he published Hypothesis Physica Nova . In this work he claimed, as had Kepler, that movement depends on the action of a spirit. He communicated with Oldenburg, the secretary of the Royal Society of London, and dedicated some of his scientific works to the Royal Society and the Paris Academy. Leibniz was also in contact with Carcavi, the Royal Librarian in Paris. As Ross explains in :-
Although Leibniz's interests were clearly developing in a scientific direction, he still hankered after a literary career. All his life he prided himself on his poetry (mostly Latin), and boasted that he could recite the bulk of Virgil's "Aeneid" by heart. During this time with Boineburg he would have passed for a typical late Renaissance humanist.
Leibniz wished to visit Paris to make more scientific contacts. He had begun construction of a calculating machine which he hoped would be of interest. He formed a political plan to try to persuade the French to attack Egypt and this proved the means of his visiting Paris. In 1672 Leibniz went to Paris on behalf of Boineburg to try to use his plan to divert Louis XIV from attacking German areas. His first object in Paris was to make contact with the French government but, while waiting for such an opportunity, Leibniz made contact with mathematicians and philosophers there, in particular Arnauld and Malebranche, discussing with Arnauld a variety of topics but particularly church reunification.

In Paris Leibniz studied mathematics and physics under Christiaan Huygens beginning in the autumn of 1672. On Huygens' advice, Leibniz read Saint-Vincent's work on summing series and made some discoveries of his own in this area. Also in the autumn of 1672, Boineburg's son was sent to Paris to study under Leibniz which meant that his financial support was secure. Accompanying Boineburg's son was Boineburg's nephew on a diplomatic mission to try to persuade Louis XIV to set up a peace congress. Boineburg died on 15 December but Leibniz continued to be supported by the Boineburg family.

In January 1673 Leibniz and Boineburg's nephew went to England to try the same peace mission, the French one having failed. Leibniz visited the Royal Society, and demonstrated his incomplete calculating machine. He also talked with Hooke, Boyle and Pell. While explaining his results on series to Pell, he was told that these were to be found in a book by Mouton. The next day he consulted Mouton's book and found that Pell was correct. At the meeting of the Royal Society on 15 February, which Leibniz did not attend, Hooke made some unfavourable comments on Leibniz's calculating machine. Leibniz returned to Paris on hearing that the Elector of Mainz had died. Leibniz realised that his knowledge of mathematics was less than he would have liked so he redoubled his efforts on the subject.

The Royal Society of London elected Leibniz a fellow on 19 April 1673. Leibniz met Ozanam and solved one of his problems. He also met again with Huygens who gave him a reading list including works by Pascal, Fabri, Gregory, Saint-Vincent, Descartes and Sluze. He began to study the geometry of infinitesimals and wrote to Oldenburg at the Royal Society in 1674. Oldenburg replied that Newton and Gregory had found general methods. Leibniz was, however, not in the best of favours with the Royal Society since he had not kept his promise of finishing his mechanical calculating machine. Nor was Oldenburg to know that Leibniz had changed from the rather ordinary mathematician who visited London, into a creative mathematical genius. In August 1675 Tschirnhaus arrived in Paris and he formed a close friendship with Leibniz which proved very mathematically profitable to both.

It was during this period in Paris that Leibniz developed the basic features of his version of the calculus. In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar for both integral and fractional .

Newton wrote a letter to Leibniz, through Oldenburg, which took some time to reach him. The letter listed many of Newton's results but it did not describe his methods. Leibniz replied immediately but Newton, not realising that his letter had taken a long time to reach Leibniz, thought he had had six weeks to work on his reply. Certainly one of the consequences of Newton's letter was that Leibniz realised he must quickly publish a fuller account of his own methods.

Newton wrote a second letter to Leibniz on 24 October 1676 which did not reach Leibniz until June 1677 by which time Leibniz was in Hanover. This second letter, although polite in tone, was clearly written by Newton believing that Leibniz had stolen his methods. In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.

Newton was to claim, with justification, that
... not a single previously unsolved problem was solved ...
by Leibniz's approach but the formalism was to prove vital in the latter development of the calculus. Leibniz never thought of the derivative as a limit. This does not appear until the work of d'Alembert.

Leibniz would have liked to have remained in Paris in the Academy of Sciences, but it was considered that there were already enough foreigners there and so no invitation came. Reluctantly Leibniz accepted a position from the Duke of Hanover, Johann Friedrich, of librarian and of Court Councillor at Hanover. He left Paris in October 1676 making the journey to Hanover via London and Holland. The rest of Leibniz's life, from December 1676 until his death, was spent at Hanover except for the many travels that he made.

His duties at Hanover :-
... as librarian were onerous, but fairly mundane: general administration, purchase of new books and second-hand libraries, and conventional cataloguing.
He undertook a whole collection of other projects however. For example one major project begun in 1678-79 involved draining water from the mines in the Harz mountains. His idea was to use wind power and water power to operate pumps. He designed many different types of windmills, pumps, gears but :-
... every one of these projects ended in failure. Leibniz himself believed that this was because of deliberate obstruction by administrators and technicians, and the workers' fear that technological progress would cost them their jobs.
In 1680 Duke Johann Friedrich died and his brother Ernst August became the new Duke. The Harz project had always been difficult and it failed by 1684. However Leibniz had achieved important scientific results becoming one of the first people to study geology through the observations he compiled for the Harz project. During this work he formed the hypothesis that the Earth was at first molten.

Another of Leibniz's great achievements in mathematics was his development of the binary system of arithmetic. He perfected his system by 1679 but he did not publish anything until 1701 when he sent the paper Essay d'une nouvelle science des nombres to the Paris Academy to mark his election to the Academy. Another major mathematical work by Leibniz was his work on determinants which arose from his developing methods to solve systems of linear equations. Although he never published this work in his lifetime, he developed many different approaches to the topic with many different notations being tried out to find the one which was most useful. An unpublished paper dated 22 January 1684 contains very satisfactory notation and results.

Leibniz continued to perfect his metaphysical system in the 1680s attempting to reduce reasoning to an algebra of thought. Leibniz published Meditationes de Cognitione, Veritate et Ideis which clarified his theory of knowledge. In February 1686, Leibniz wrote his Discours de métaphysique .

Another major project which Leibniz undertook, this time for Duke Ernst August, was writing the history of the Guelf family, of which the House of Brunswick was a part. He made a lengthy trip to search archives for material on which to base this history, visiting Bavaria, Austria and Italy between November 1687 and June 1690. As always Leibniz took the opportunity to meet with scholars of many different subjects on these journeys. In Florence, for example, he discussed mathematics with Viviani who had been Galileo's last pupil. Although Leibniz published nine large volumes of archival material on the history of the Guelf family, he never wrote the work that was commissioned.

In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus... in Acta Eruditorum, a journal established in Leipzig two years earlier. The paper contained the familiar d notation, the rules for computing the derivatives of powers, products and quotients. However it contained no proofs and Jacob Bernoulli called it an enigma rather than an explanation.

In 1686 Leibniz published, in Acta Eruditorum, a paper dealing with the integral calculus with the first appearance in print of the ∫ notation.

Newton's Principia appeared the following year. Newton's 'method of fluxions' was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736. This time delay in the publication of Newton's work resulted in a dispute with Leibniz.

Another important piece of mathematical work undertaken by Leibniz was his work on dynamics. He criticised Descartes' ideas of mechanics and examined what are effectively kinetic energy, potential energy and momentum. This work was begun in 1676 but he returned to it at various times, in particular while he was in Rome in 1689. It is clear that while he was in Rome, in addition to working in the Vatican library, Leibniz worked with members of the Accademia. He was elected a member of the Accademia at this time. Also while in Rome he read Newton's Principia. His two part treatise Dynamica studied abstract dynamics and concrete dynamics and is written in a somewhat similar style to Newton's Principia. Ross writes in :-
... although Leibniz was ahead of his time in aiming at a genuine dynamics, it was this very ambition that prevented him from matching the achievement of his rival Newton. ... It was only by simplifying the issues... that Newton succeeded in reducing them to manageable proportions.
Leibniz put much energy into promoting scientific societies. He was involved in moves to set up academies in Berlin, Dresden, Vienna, and St Petersburg. He began a campaign for an academy in Berlin in 1695, he visited Berlin in 1698 as part of his efforts and on another visit in 1700 he finally persuaded Friedrich to found the Brandenburg Society of Sciences on 11 July. Leibniz was appointed its first president, this being an appointment for life. However, the Academy was not particularly successful and only one volume of the proceedings were ever published. It did lead to the creation of the Berlin Academy some years later.

Other attempts by Leibniz to found academies were less successful. He was appointed as Director of a proposed Vienna Academy in 1712 but Leibniz died before the Academy was created. Similarly he did much of the work to prompt the setting up of the St Petersburg Academy, but again it did not come into existence until after his death.

It is no exaggeration to say that Leibniz corresponded with most of the scholars in Europe. He had over 600 correspondents. Among the mathematicians with whom he corresponded was Grandi. The correspondence started in 1703, and later concerned the results obtained by putting into . Leibniz also corresponded with Varignon on this paradox. Leibniz discussed logarithms of negative numbers with Johann Bernoulli, see .

In 1710 Leibniz published Théodicée a philosophical work intended to tackle the problem of evil in a world created by a good God. Leibniz claims that the universe had to be imperfect, otherwise it would not be distinct from God. He then claims that the universe is the best possible without being perfect. Leibniz is aware that this argument looks unlikely - surely a universe in which nobody is killed by floods is better than the present one, but still not perfect. His argument here is that the elimination of natural disasters, for example, would involve such changes to the laws of science that the world would be worse. In 1714 Leibniz wrote Monadologia which synthesised the philosophy of his earlier work, the Théodicée .

Much of the mathematical activity of Leibniz's last years involved the priority dispute over the invention of the calculus. In 1711 he read the paper by Keill in the Transactions of the Royal Society of London which accused Leibniz of plagiarism. Leibniz demanded a retraction saying that he had never heard of the calculus of fluxions until he had read the works of Wallis. Keill replied to Leibniz saying that the two letters from Newton, sent through Oldenburg, had given:-
... pretty plain indications... whence Leibniz derived the principles of that calculus or at least could have derived them.
Leibniz wrote again to the Royal Society asking them to correct the wrong done to him by Keill's claims. In response to this letter the Royal Society set up a committee to pronounce on the priority dispute. It was totally biased, not asking Leibniz to give his version of the events. The report of the committee, finding in favour of Newton, was written by Newton himself and published as Commercium epistolicum near the beginning of 1713 but not seen by Leibniz until the autumn of 1714. He learnt of its contents in 1713 in a letter from Johann Bernoulli, reporting on the copy of the work brought from Paris by his nephew . Leibniz published an anonymous pamphlet Charta volans setting out his side in which a mistake by Newton in his understanding of second and higher derivatives, spotted by Johann Bernoulli, is used as evidence of Leibniz's case.

The argument continued with Keill who published a reply to Charta volans. Leibniz refused to carry on the argument with Keill, saying that he could not reply to an idiot. However, when Newton wrote to him directly, Leibniz did reply and gave a detailed description of his discovery of the differential calculus. From 1715 up until his death Leibniz corresponded with Samuel Clarke, a supporter of Newton, on time, space, freewill, gravitational attraction across a void and other topics, see , , and .

In Leibniz is described as follows:-
Leibniz was a man of medium height with a stoop, broad-shouldered but bandy-legged, as capable of thinking for several days sitting in the same chair as of travelling the roads of Europe summer and winter. He was an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation.
Ross, in , points out that Leibniz's legacy may have not been quite what he had hoped for:-
It is ironical that one so devoted to the cause of mutual understanding should have succeeded only in adding to intellectual chauvinism and dogmatism. There is a similar irony in the fact that he was one of the last great polymaths - not in the frivolous sense of having a wide general knowledge, but in the deeper sense of one who is a citizen of the whole world of intellectual inquiry. He deliberately ignored boundaries between disciplines, and lack of qualifications never deterred him from contributing fresh insights to established specialisms. Indeed, one of the reasons why he was so hostile to universities as institutions was because their faculty structure prevented the cross-fertilisation of ideas which he saw as essential to the advance of knowledge and of wisdom. The irony is that he was himself instrumental in bringing about an era of far greater intellectual and scientific specialism, as technical advances pushed more and more disciplines out of the reach of the intelligent layman and amateur.

Written by J J O'Connor and E F Robertson
Last Update October 1998

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Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

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Who introduced the technique of function notation?

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Who introduced the technique of function notation?

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Biography

Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister. Their son Leonhard Euler was born in Basel, but the family moved to Riehen when he was one year old and it was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we have mentioned, some mathematical training and he was able to teach his son elementary mathematics along with other subjects.

Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school. However his interest in mathematics had certainly been sparked by his father's teaching, and he read mathematics texts on his own and took some private lessons. Euler's father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered. Euler's own account given in his unpublished autobiographical writings, see , is as follows:-

... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

In 1723 Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton. He began his study of theology in the autumn of 1723, following his father's wishes, but, although he was to be a devout Christian all his life, he could not find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Euler obtained his father's consent to change to mathematics after Johann Bernoulli had used his persuasion. The fact that Euler's father had been a friend of Johann Bernoulli's in their undergraduate days undoubtedly made the task of persuasion much easier.

Euler completed his studies at the University of Basel in 1726. He had studied many mathematical works during his time in Basel, and Calinger has reconstructed many of the works that Euler read with the advice of Johann Bernoulli. They include works by Varignon, Descartes, Newton, Galileo, van Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already a paper in print, a short article on isochronous curves in a resisting medium. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship.

The Prize of 1727 went to Bouguer, an expert on mathematics relating to ships, but Euler's essay won him second place which was a fine achievement for the young graduate. However, Euler now had to find himself an academic appointment and when died in St Petersburg in July 1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology. He accepted the post in November 1726 but stated that he did not want to travel to Russia until the spring of the following year. He had two reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance of a post at the University of Basel since the professor of physics there had died. Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair. Almost certainly his youth (he was 19 at the time) was against him. However Calinger suggests:-

This decision ultimately benefited Euler, because it forced him to move from a small republic into a setting more adequate for his brilliant research and technological work.

As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April 1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lübeck arriving in St Petersburg on 17 May 1727. He had joined the St Petersburg Academy of Sciences two years after it had been founded by Catherine I the wife of Peter the Great. Through the requests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. At St Petersburg Euler had many colleagues who would provide an exceptional environment for him :-

Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.

Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. In St Petersburg he lived with Daniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee, brandy and other delicacies from Switzerland. Euler became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post.

Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of mathematics. The financial improvement which came from this appointment allowed Euler to marry which he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet.

We will examine Euler's mathematical achievements later in this article but at this stage it is worth summarising Euler's work in this period of his career. This is done in as follows:-

... after 1730 he carried out state projects dealing with cartography, science education, magnetism, fire engines, machines, and ship building. ... The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics. He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.

The publication of many articles and his book Mechanica (1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work.

Euler's health problems began in 1735 when he had a severe fever and almost lost his life. However, he kept this news from his parents and members of the Bernoulli family back in Basel until he had recovered. In his autobiographical writings Euler says that his eyesight problems began in 1738 with overstrain due to his cartographic work and that by 1740 he had :-

... lost an eye and [the other] currently may be in the same danger.

However, Calinger in argues that Euler's eyesight problems almost certainly started earlier and that the severe fever of 1735 was a symptom of the eyestrain. He also argues that a portrait of Euler from 1753 suggests that by that stage the sight of his left eye was still good while that of his right eye was poor but not completely blind. Calinger suggests that Euler's left eye became blind from a later cataract rather than eyestrain.

By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740. On both occasions he shared the first prize with others. Euler's reputation was to bring an offer to go to Berlin, but at first he preferred to remain in St Petersburg. However political turmoil in Russia made the position of foreigners particularly difficult and contributed to Euler changing his mind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin where an Academy of Science was planned to replace the Society of Sciences. He left St Petersburg on 19 June 1741, arriving in Berlin on 25 July. In a letter to a friend Euler wrote:-

I can do just what I wish [in my research] ... The king calls me his professor, and I think I am the happiest man in the world.

Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration he bought books and instruments for the St Petersburg Academy, he continued to write scientific reports for them, and he educated young Russians.

Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as director of mathematics. He deputised for Maupertuis in his absence and the two became great friends. Euler undertook an unbelievable amount of work for the Academy :-

... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.

This was not the limit of his duties by any means. He served on the committee of the Academy dealing with the library and of scientific publications. He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal.

During the twenty-five years spent in Berlin, Euler wrote around 380 articles. He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics (extending the book by Robins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publication Letters to a Princess of Germany (3 vols., 1768-72).

In 1759 Maupertuis died and Euler assumed the leadership of the Berlin Academy, although not the title of President. The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour. Euler, who had argued with d'Alembert on scientific matters, was disturbed when Frederick offered d'Alembert the presidency of the Academy in 1763. However d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave.

In 1766 Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soon after his return to Russia, Euler became almost entirely blind after an illness. In 1771 his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the fire, still in 1771, restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind. Because of his remarkable memory he was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his return to St Petersburg (when Euler was 59) he produced almost half his total works despite the total blindness.

Euler of course did not achieve this remarkable level of output without help. He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St Petersburg in 1766 (becoming its secretary in 1769) and Christoph Euler who had a military career. Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772. Fuss, who was Euler's grandson-in-law, became his assistant in 1776. Yushkevich writes in :-

.. the scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculating tables, and sometimes compiled examples.

For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on the motion of the moon, published in 1772. Fuss helped Euler prepare over 250 articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in 1776.

He also wrote a eulogy of Euler, which you can see at THIS LINK.

Yushkevich describes the day of Euler's death in :-

On 18 September 1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening.

After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years.

Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done.

He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765).

Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic. Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers were always prime if is a power of 2. Euler verified this for = 1, 2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case is divisible by 641 and so is not prime. Euler also studied other unproved results of Fermat and in so doing introduced the Euler phi function , the number of integers with and coprime to . He proved another of Fermat's assertions, namely that if and are coprime then has no divisor of the form , in 1749.

Here the sum is over all natural numbers while the product is over all prime numbers.

By 1739 Euler had found the rational coefficients in in terms of the Bernoulli numbers.

Other work done by Euler on infinite series included the introduction of his famous Euler's constant γ, in 1735, which he showed to be the limit of

as tends to infinity. He calculated the constant γ to 16 decimal places. Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result

in a letter to Goldbach. Like most of Euler's work there was a fair time delay before the results were published; this result was not published until 1755.

Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series. In particular he wrote :-

Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.

He then goes on to describe what is now called the Euler- Maclaurin summation formula. Two years later Stirling replied telling Euler that Maclaurin:-

... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.

Euler replied:-

... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.

Some of Euler's number theory results have been mentioned above. Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of . Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form for integers and . Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring.

One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysin infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions. This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously. Also in this work Euler gave the formula

In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive values although he had discovered the formula

in 1727. He published his full theory of logarithms of complex numbers in 1751.

Analytic functions of a complex variable were investigated by Euler in a number of different contexts, including the study of orthogonal trajectories and cartography. He discovered the Cauchy- Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics.

In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus of finite differences. The work makes a thorough investigation of how differentiation behaves under substitutions.

In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions. He also studied beta and gamma functions, which he had introduced first in 1729. Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively. As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work.

The calculus of variations is another area in which Euler made fundamental discoveries. His work Methodus inveniendi lineas curvas ... published in 1740 began the proper study of the calculus of variations. In it is noted that Carathéodory considered this as:-

... one of the most beautiful mathematical works ever written.

Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.

Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces. Many unpublished results by Euler in this area were rediscovered by Gauss. Other geometric investigations led him to fundamental ideas in topology such as the Euler characteristic of a polyhedron.

In 1736 Euler published Mechanica which provided a major advance in mechanics. As Yushkevich writes in :-

The distinguishing feature of Euler's investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.

In Mechanica Euler considered the motion of a point mass both in a vacuum and in a resisting medium. He analysed the motion of a point mass under a central force and also considered the motion of a point mass on a surface. In this latter topic he had to solve various problems of differential geometry and geodesics.

Mechanica was followed by another important work in rational mechanics, this time Euler's two volume work on naval science. It is described in as:-

Outstanding in both theoretical and applied mechanics, it addresses Euler's intense occupation with the problem of ship propulsion. It applies variational principles to determine the optimal ship design and first established the principles of hydrostatics ... Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.

Of course hydrostatics had been studied since Archimedes, but Euler gave a definitive version.

In 1765 Euler published another major work on mechanics Theoria motus corporum solidorum Ⓣ(Theory of the motion of solid bodies) in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. He considered the Euler angles and studied rotational problems which were motivated by the problem of the precession of the equinoxes.

Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces of work through the 1750s setting up the main formulae for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid. In 1752 he wrote:-

However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ...

Euler contributed to knowledge in many other areas, and in all of them he employed his mathematical knowledge and skill. He did important work in astronomy including :-

... determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, consideration of the physical nature of comets, .... His most outstanding works, for which he won many prizes from the Paris Académie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time.

In fact Euler's lunar theory was used by Tobias Mayer in constructing his tables of the moon. In 1765 Mayer's widow received £3000 from Britain for the contribution the tables made to the problem of the determination of the longitude, while Euler received £300 from the British government for his theoretical contribution to the work.

Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739 in which he tried to make music:-

... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing.

However, according to the work was:-

... for musicians too advanced in its mathematics and for mathematicians too musical.

Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and it appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography.

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Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister.

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Earliest Known Uses of the word "Domain" by Arthur Cayley 

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Earliest Known Uses of the word "Domain" by Arthur Cayley 

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Arthur Cayley Bio: https://mathshistory.st-andrews.ac.uk/Biographies/Cayley/

** Click on photo for a little background on the term Domain

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Earliest Known Uses of the word "Range" by John Spare

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Earliest Known Uses of the word "Range" by John Spare

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** Click on the photo for a little background on the term Range