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What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational

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Classification of Numbers (Natural, Whole, Integers, Rational, Irrational, Real) - Nerdstudy

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Irrational Numbers, Rational Numbers: with real world examples

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Rational and irrational numbers definitions, examples, a video about ratios, and more.

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Rational Number -- from Wolfram MathWorld

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It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.

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Distinguish between rational and irrational numbers

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In this lesson you will learn how to tell the difference between rational and irrational numbers by looking for repeating decimals.

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Check out the subsets of the Real Numbers shown in the diagram at the right. Notice that the rational and irrational numbers are contained within the set of Real Numbers.

A rational number is a number that can be expressed as a fraction (ratio) in the form where p and q are integers and q is not zero.

Examples: (fraction). A ratio nal number can be expressed as a ratio

When a rational number fraction is divided to form a decimal value,
it becomes a terminating or repeating decimal.

Some rational fractions may produce a large number of digits in their repeating patterns, which may exceed the size of the viewing screen on a calculator. The fraction 53/83 has a calculator display of 0.6385542169, which shows no repeating pattern, when in reality the pattern will repeat after 41 digits.

To convert a repeating decimal to a fraction:

To show that the rational numbers are dense:
(between any two rationals there is another rational)

An irrational number is a number that is NOT rational. It cannot be expressed as a fraction with integer values in the numerator and denominator.

Examples:

While it is popular to use 3.14 or to represent "pi", these values are only estimates or approximations. Notice the differences in the decimal representations on the calculator screen at the right.

Properties of Rational and Irrational Numbers:

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Rational and Irrational Numbers - MathBitsNotebook(A1 - CCSS Math)

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Check out the subsets of the Real Numbers shown in the diagram at the right. Notice that the rational and irrational numbers are contained within the set of Real Numbers. A rational number is a number that can be expressed as a fraction (ratio) in the form where p and q are integers and q is not zero.

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Providing instructional and assessment tasks, lesson plans, and other resources for teachers, assessment writers, and curriculum developers since 2011.

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IXL | Identify rational and irrational numbers | 8th grade math

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Improve your math knowledge with free questions in “Identify rational and irrational numbers” and thousands of other math skills.

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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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Rational and irrational numbers definitions, examples, a video about ratios, and more.