Rational vs irrational How do we know whats

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Rational vs. irrational How do we know what’s what?

Rational vs. irrational How do we know what’s what?

What is a rational number? �A rational number is a number that can be

What is a rational number? �A rational number is a number that can be written as a simple fraction ◦ Ex. 5 5/1 � ◦ Ex. 0. 3333333… ◦ Ex. 1. 32567… Formal definition: 1/3� ? /? x “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?

Are these rational numbers? P Q p/q = yes or no 1 1 1/1

Are these rational numbers? 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. 414213… No When looking at the decimals, what can we conclude about determining if it is a rational number? The decimal has to REPEAT or TERMINATE!

Story of Irrational Numbers The ancient Greek mathematician Pythagoras believed that all numbers were

Story of Irrational Numbers 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,

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!

What is an Irrational Number? �An Irrational Number is a real number that cannot

What is an Irrational Number? �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!

Label as Rational or Irrational � 1/4 � 2. 3333…. � -45. 32 �

Label as Rational or Irrational � 1/4 � 2. 3333…. � -45. 32 � 7 � 0. 873250 � 9. 914701…

Number Systems Rational Numbers Integers Whole Numbers Counting # Irrational Numbers

Number Systems Rational Numbers Integers Whole Numbers Counting # Irrational Numbers

Rational Numbers � Any Number that can be written as a fraction. � Terminating

Rational 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. 252525….

Integers � Positive and negative whole numbers � No fractions � No decimals �

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

Whole Numbers � No fractions � Not decimals � No negatives � Examples: 0,

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

Natural numbers � Aka counting numbers � Does not include zero � No negatives

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

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

Irrational numbers � Any number that cannot be written as a fraction p/q � Non- repeating decimals � Non- terminating decimals � Examples: 1. 34567231. . , √ 2, π

What kind of number is this? � Name the numbers systems that each of

What kind of number is this? � Name the numbers systems that each of the following numbers belongs to. -3 2/3 Integers, Rational numbers 0. 324… Irrational numbers 23 Counting numbers, Whole numbers, Integers, Rational numbers

Review: How to convert fractions to decimals � It’s Simple! � Divide the numerator

Review: How to convert fractions to decimals � 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 decimals to fractions � Change the decimal to a fraction

Review: How to convert decimals to fractions � 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