Rational vs. irrationalHow do we know what’s what?

A rational number is a number that can be written as a simple fraction◦Ex. 5 5/1 ◦ Ex. 0.3333333… 1/3◦Ex. 1.32567… ?/? x

Formal definition:

“A rational number is a number that can be written in the form p/q where p and q are integers and q is not equal to zero”

* Why can’t q be zero?

What is a rational number?

P Q p/q = yes or no

1 1 1/1 1 Yes

1 6 1/6 1.66666… Yes

55 100 55/100 0.55 Yes

253 10 253/10 25.3 Yes

7 0 7/0 ? No

√9 1 3/1 3 Yes

√2 1 √2 1.414213… No

Are these rational numbers?

When looking at the decimals, what can we conclude about determining if it is a rational number?

The decimal has to REPEAT or TERMINATE!

The ancient Greek mathematician Pythagoras believed that all

numbers were rational (could be written as a fraction), but one of

his students Hippasus proved (using geometry, it is thought)

that you could not represent the square root of 2 as a fraction, and

so it was irrational.

Story of Irrational Numbers

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect

values. But he could not disprove Hippasus' "irrational numbers theory" and so Hippasus was

thrown overboard and drowned!

Story of Irrational Numbers

An Irrational Number is a real number that cannot be written as a simple fraction.◦Irrational means not Rational

Ex. Π = 3.14159… Not Rational! Ex. √3= 1.73205080… Not Rational!

*Notice: These decimals cannot be written as a fraction!

What is an Irrational Number?

1/4

2.3333….

-45.32

7

0.873250

9.914701…

Label as Rational or Irrational

Number Systems

Rational Numbers

Integers

Whole Numbers

Counting #

Irrational Numbers

Any Number that can be written as a fraction.

Terminating or repeating decimals Fractions Integers Can be positive or negative Examples: 0.33333…, 0.425, ½ ,- ¾ , -2,

3.25, 0.2525252525….

Rational Numbers

Positive and negative whole numbers No fractions No decimals Zero IS included! examples: …-3,-2,-1,0,1,2,3,…..

Integers

No fractions Not decimals No negatives Examples: 0,1,2,3,4,5,…..

Whole Numbers

Aka counting numbers Does not include zero No negatives No fractions No decimals Examples: 1,2,3,4,5,…..

Natural numbers

Any number that cannot be written as a fraction p/q

Non- repeating decimals Non- terminating decimals Examples: 1.34567231...., √2, π

Irrational numbers

Name the numbers systems that each of the following numbers belongs to.

-3

2/3

0.324…

23

What kind of number is this?

Integers, Rational numbers

Rational numbers

Irrational numbers

Counting numbers, Whole numbers, Integers, Rational numbers

It’s Simple! Divide the numerator by the denominator Ex 3/7 = 3÷ 7 = 0.42857143 Round to the nearest hundredth = 0.43

Review: How to convert fractions to decimals

Change the decimal to a fraction by using the given digits as the numerator and the place value as the denominator

0.125 = 125 / 1,000 Then we reduce the fraction!

125/1,000 = 1/8

Review: How to convert decimals to fractions