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How are irrational numbers approximated?

How can we compare a rational and irrational
number?

How can one determine the approximate the value of an irrational number?

What is the Pythagorean Theorem and how is it
applied?

How does the Pythagorean Theorem help you
build a bike ramp?

How does the Pythagorean Theorem help you to
find the distances between points on a coordinate plane?

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Students will use their knowledge of square roots to learn how to find the square roots of perfect
squares.

Students will use their knowledge of perfect squares and ordering to estimate the square root of a number that is not a perfect square to one decimal place.

Students will be able to estimate the value of expressions that include the square root of a number
that is not a perfect square.

Students will estimate expressions involving a square root, and place them on a number line.

Students will use their knowledge of number sense of understanding of the number system to include rational, irrational, and real numbers.

Students will determine if a number is rational or irrational.

Students will determine which set of numbers a given number belongs to. For example the number 3
is a real, rational, whole, and natural number, but the number 1/3 is only real and rational.

Students will learn what a perfect cube is and be able to find the cube root of a small perfect cube.

Students understand informally that every number has a decimal expansion.

For rational numbers, students show that the decimal expansion repeats eventually, and convert a
decimal expansion which repeats eventually into a rational number.

Students can convert a decimal expansion that repeats into a rational number.

Students will use square roots to find the radius of a cylinder or cone when given the height and volume.

Students will use cube roots to find the radius of a sphere when given the volume.

Students will use their knowledge of a right triangle to find the third side of a triangle using Pythagorean Theorem.

Students will use their knowledge of a right triangle to find the hypotenuse of a triangle using Pythagorean Theorem.

Students will use their knowledge of a right triangle to find the length of a leg of a triangle using Pythagorean Theorem.

Students will use their knowledge of Pythagorean Theorem using the converse to prove that the triangle has a right angle.

Students will use their knowledge of Pythagorean Theorem to solve word problems.

Students will use their knowledge of Pythagorean Theorem to find the missing length of a 2dimensional figure and a 3 dimensional figure.

Students will use their knowledge of Pythagorean Theorem to explain its proof and converse of the theorem.

Students will be able to explain how the distance formula follows from the Pythagorean Theorem.

Students will use the distance formula to find the distance from a set of two ordered pairs.

Students will use the distance formula to find the distance when given two points on a
coordinate plane.

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PRIORITY:
NS.2​ Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g.,π 2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

G.7​ Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

G.8​ Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

SUPPORTING:
NS.1​ Know that numbers that are not rational are called irrational. Understand informally that every
number has a decimal expansion; for rational numbers show that the decimal expansion repeats
eventually, and convert a decimal expansion which repeats eventually into a rational number.

EE.2​ Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

G.6​ Explain a proof of the Pythagorean Theorem and its converse.

G.9​ Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.

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Categorize
Sort
Simplify
Estimate
Evaluate
Approximate
Apply
Validate
Investigate
Utilize
Subscript
Proof
Apply
Converse

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Rational Numbers
Root
Cubic
Square Root
Cube Root
Irrational Numbers
Radicals
Pythagorean Theorem
Theorem
Pythagorean Triplets
Distance Formula
Converse

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How to Use the Pythagorean Theorem. Step By Step Examples and Practice

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When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above . Look at the following examples to see pictures of the formula.

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Use Pythagorean theorem to find right triangle side lengths

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Use Pythagorean theorem to find right triangle side lengths (practice) | Khan Academy

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Pythagorean theorem word problems

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Pythagorean theorem challenge (practice) | Khan Academy

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Distance between two points

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Use the Pythagorean theorem to find the distance between two points on the coordinate plane.

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Right triangles and the Pythagorean Theorem

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So let's talk a little bit about rational numbers. And the simple way to think about it is any number that can be represented as the ratio of two integers is a rational number. So for example, any integer is a rational number. 1 can be represented as 1/1 or as negative 2 over negative 2 or as 10,000/10,000. In all of these cases, these are all different representations of the number 1, ratio of two integers. And I obviously can have an infinite number of representations of 1 in this way, the same number over the same number. The number negative 7 could be represented as negative 7/1, or 7 over negative 1, or negative 14 over positive 2. And I could go on, and on, and on, and on. So negative 7 is definitely a rational number. It can be represented as the ratio of two integers. But what about things that are not integers? For example, let us imagine-- oh, I don't know-- 3.75. How can we represent that as the ratio of two integers? Well, 3.75, you could rewrite that as 375/100, which is the same thing as 750/200. Or you could say, hey, 3.75 is the same thing as 3 and 3/4-- so let me write it here-- which is the same thing as-- that's 15/4. 4 times 3 is 12, plus 3 is 15, so you could write this. This is the same thing as 15/4. Or we could write this as negative 30 over negative 8. I just multiplied the numerator and the denominator here by negative 2. But just to be clear, this is clearly rational. I'm giving you multiple examples of how this can be represented as the ratio of two integers. Now, what about repeating decimals? Well, let's take maybe the most famous of the repeating decimals. Let's say you have 0.333, just keeps going on and on forever, which we can denote by putting that little bar on top of the 3. This is 0.3 repeating. And we've seen-- and later we'll show how you can convert any repeating decimal as the ratio of two integers-- this is clearly 1/3. Or maybe you've seen things like 0.6 repeating, which is 2/3. And there's many, many, many other examples of this. And we'll see any repeating decimal, not just one digit repeating. Even if it has a million digits repeating, as long as the pattern starts to repeat itself over and over and over again, you can always represent that as the ratio of two integers. So I know what you're probably thinking. Hey, Sal, you've just included a lot. You've included all of the integers. You've included all of finite non-repeating decimals, and you've also included repeating decimals. What is left? Are there any numbers that are not rational? And you're probably guessing that there are, otherwise people wouldn't have taken the trouble of trying to label these as rational. And it turns out-- as you can imagine-- that actually some of the most famous numbers in all of mathematics are not rational. And we call these numbers irrational numbers. And I've listed there just a few of the most noteworthy examples. Pi-- the ratio of the circumference to the diameter of a circle-- is an irrational number. It never terminates. It goes on and on and on forever, and it never repeats. e, same thing-- never terminates, never repeats. It comes out of continuously compounding interest. It comes out of complex analysis. e shows up all over the place. Square root of 2, irrational number. Phi, the golden ratio, irrational number. So these things that really just pop out of nature, many of these numbers are irrational. Now, you might say, OK, are these irrational? These are just these special kind of numbers. But maybe most numbers are rational, and Sal's just picked out some special cases here. But the important thing to realize is they do seem exotic, and they are exotic in certain ways. But they aren't uncommon. It actually turns out that there is always an irrational number between any two rational numbers. Well, we could go on and on. There's actually an infinite number. But there's at least one, so that gives you an idea that you can't really say that there are fewer irrational numbers than rational numbers. And in a future video, we'll prove that you give me two rational numbers-- rational 1, rational 2-- there's going to be at least one irrational number between those, which is a neat result, because irrational numbers seem to be exotic. Another way to think about it-- I took the square root of 2, but you take the square root of any non-perfect square, you're going to end up with an irrational number. You take the sum of an irrational and a rational number-- and we'll see this later on. We'll prove it to ourselves. The sum of an irrational and a rational is going to be irrational. The product of an irrational and a rational is going to be irrational. So there's a lot, a lot, a lot of irrational numbers out there.

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Intro to rational & irrational numbers | Algebra (video) | Khan Academy

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So let's talk a little bit about rational numbers. And the simple way to think about it is any number that can be represented as the ratio of two integers is a rational number. So for example, any integer is a rational number.

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Distance Formula

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Distance between two points | Analytic geometry (practice) | Khan Academy

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Use the Pythagorean theorem to find the distance between two points on the coordinate plane.

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Distance formula

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What Is the Distance Formula? | Virtual Nerd

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Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

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Irrational Numbers

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Math and Brain Games. Mathematics Question Database. For K-12 kids, teachers and parents.

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Coordinates and Distance formula

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Coordinates And Distance Formula by Shmoop

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Irrational Numbers

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An Irrational Number is a real number that cannot be written as a simple fraction:

Irrational means not Rational (no ratio)

Let's look at what makes a number rational or irrational ...

Rational Numbers

A Number can be written as a Ratio of two integers (ie a simple fraction).

Example: 1.5 is rational, because it can be written as the ratio 3/2

Example: 7 is rational, because it can be written as the ratio 7/1

Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

Irrational Numbers

But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers.
π = 3.1415926535897932384626433832795... (and more)
We cannot write down a simple fraction that equals Pi.

The popular approximation of = 3.1428571428571... is close but not accurate.

Another clue is that the decimal goes on forever without repeating.

Cannot Be Written as a Fraction
It is irrational because it cannot be written as a ratio (or fraction),
not because it is crazy!
So we can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

Example: 9.5 can be written as a simple fraction like this:

So it is a rational number (and so is not irrational)

Here are some more examples:

Square Root of 2

Let's look at the square root of 2 more closely.

When we draw a square of size "1",
what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc)

But it is not a number like 3, or five-thirds, or anything like that ...

Famous Irrational Numbers

Pi is a famous irrational number. People have calculated Pi to over a quadrillion decimal places and still there is no pattern. The first few digits look like this:

3.1415926535897932384626433832795 (and more ...)

The number e ( Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this:

2.7182818284590452353602874713527 (and more ...)

The Golden Ratio is an irrational number. The first few digits look like this:

1.61803398874989484820... (and more ...)

Many square roots, cube roots, etc are also irrational numbers. Examples:

√3

1.7320508075688772935274463415059 (etc)

√99

9.9498743710661995473447982100121 (etc)

But √4 = 2 is rational, and √9 = 3 is rational ...

... so not all roots are irrational.

Note on Multiplying Irrational Numbers

Have a look at this:

So be careful ... multiplying irrational numbers might result in a rational number!

Fun Facts ....

Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to write the square root of 2 as a fraction (using geometry, it is thought). Instead he proved the square root of 2 could not be written as a fraction, so it is irrational.

But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!

434,435,1064,2022,3987,1065,3988,2023,2990,2991

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An Irrational Number is a real number that cannot be written as a simple fraction: Irrational means not Rational (no ratio) Let's look at what makes a number rational or irrational ... A Number can be written as a Ratio of two integers (ie a simple fraction).

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Pythagorean Theorem

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Pythagoras

Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ...... and squares are made on each of the three sides, ...... then the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

Note:

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Let's check if the areas are the same:

3 ² + 4 ² = 5 ²

Calculating this becomes:

9 + 16 = 25
It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a ² + b ² = c ²

Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

Read Builder's Mathematics to see practical uses for this.

Also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

Example: What is the diagonal distance across a square of size 1?

It works the other way around, too: when the three sides of a triangle make , then the triangle is right angled.

Example: Does this triangle have a Right Angle?

They are equal, so ...

Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Yes, it does!

So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space.

Draw a square along the hypotenuse (the longest side)

Draw the same sized square on the other side of the hypotenuse

Draw lines as shown on the animation, like this:

Cut out the shapes

Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.

Watch the animation, and pay attention when the triangles start sliding around.

You may want to watch the animation a few times to understand what is happening.

The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361

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Pythagoras Over 2000 years ago there was an amazing discovery about triangles: When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together!

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Distance Formula Lesson - Free Math Help

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The distance formula quickly calculates the distance between two points. Learn the distance formula and practice applying it with this free geometry lesson.

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Distance formula/ Pythagorean Theorem presentation

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Notes- Irrational and Rational Numbers