Understanding Rational and Irrational Numbers: Identification and Examples, Summaries of Calculus

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Rational

and

Irrational Numbers

Learning Target:

Students can identify numbers as being

rational or irrational.

All REAL numbers are

made up of

RATIONAL & IRRATIONAL

numbers…

Examples of Rational Numbers

Fractions

Terminating Decimals

(stop / end)

Repeating Decimals

(decimal goes on forever)

Whole Numbers

Perfect Squares

Rational Numbers

are “Nice & Neat!”

2

3

− 9

2

1

1

4

• 6.23 17.

7. 23 − 621. 5

1 , 267

− 6 3

9 100 400

Irrational Numbers

All numbers that are not rational are considered irrational.

An irrational number can be written as a decimal, but not as

a fraction.

Irrational numbers can NOT

be written as a fraction!!

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…

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…

The Golden Ratio is an irrational number.

The first few digits look like this:

1.61803398874989484820...

Both rational and irrational numbers show up on

the number line. This is why all “REAL” numbers

include rational and irrational numbers.

Non-Real Numbers

Imaginary Numbers:

If there is a negative inside the radical, then it is a

NON-REAL NUMBER

Radicals can’t

have a negative

number

(only +𝒙 )

Infinity:

Infinity is the idea of something that has no end.

These are ok (the negative is outside of the radical)