m15 rational and irrational numbers

8/11/2019 m15 Rational and Irrational Numbers

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Rational and irrational numbers

MTE3101 Mengenal nombor

PISMP 2013


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Definition 1: Rational number

A rational number () is any number which

can be written as:

a/b

where a and b are integers and b0.


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EXAMPLE OF rational numbers

:101;217;-1-3;1020;-36(2)BOTH numerators and denominators are

integers. On the other hand, all integers are rational

numbers, because they can be written with adenominator of 1.


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Definition 2: Irrational numbers

Irrational numbers (') are numbers that

cannot be written as a fraction with the

numerator and denominator as integers.


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Examples of irrational numbers:

These are not rational numbers, becauseeither the numerator or the denominator is

not an integer.

....,,521,,43,3,2 e


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RATIONAL OR IRRATIONAL


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Square Root of 2

If you draw a square of size

"1",what is the distance across the

diagonal?


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The answer is the square rootof 2, which is

1.4142135623730950...(etc)

2
http://www.mathsisfun.com/square-root.htmlhttp://www.mathsisfun.com/square-root.html


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Famous Irrational NumbersPiis 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 eto lots of decimal

places without any pattern showing. The

first few digits look like this:

2.7182818284590452353602874713527

(and more ...)

The Golden Ratiois 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:

31.7320508075688772935274463415059

(etc)

999.9498743710661995473447982100121

(etc)

But 4 = 2 (rational), and 9 = 3 (rational) ...

... so not allroots are irrational.
http://www.mathsisfun.com/numbers/pi.htmlhttp://www.mathsisfun.com/numbers/e-eulers-number.htmlhttp://www.mathsisfun.com/numbers/golden-ratio.htmlhttp://www.mathsisfun.com/numbers/golden-ratio.htmlhttp://www.mathsisfun.com/numbers/e-eulers-number.htmlhttp://www.mathsisfun.com/numbers/pi.html


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Multiplying Irrational Numbers

= 2is irrational

But 2 2 = 2is rational So be careful ... multiplying irrational numbers

mightresult in a rational number!


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Decimal numbers

any rational number can be written as a

decimal number

but not all decimal numbers are rational

numbers


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Decimal numbers

Types of decimal numbers are rational numbers:

1. Decimal numbers that end (or terminate).

Example, the fraction 4/10 can be written as 0,4.

2 Decimal numbers that have a repeating single digit.Example, the fraction 1/3 can be written as or asThe dot and bar notations are equivalent and bothrepresent recurring 3's, i.e. = =0.333....

3 Decimal numbers that have a recurring pattern ofmultiple digits. For example, the fraction 2/11 canalso be written as . The bar represents arecurring pattern of 1 and 8's i.e. =0,181818....

3.0 3.0

3.0 3.0

18.0

18.0


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Converting terminating decimals into

rational numbers

A decimal number has an integer part and afractional part.

For example, 10.589 has an integer part of 10 and

a fractional part of 0.589 because10+0.589=10.589.

The fractional part can be written as a rationalnumber, i.e. with a numerator and denominator

that are integers. Each digit after the decimal point is a fraction

with a denominator in increasing powers of 10.


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Example

Therefore, 10.589=10+5/10+8/100+9/1000

=10000/1000+500/1000+80/1000+9/1000

=10589/1000


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Converting recurring decimals into

rational numbers

When the decimal is a recurring decimal, a bit

more work is needed to write the fractional

part of the decimal number as a fraction.

Eg: 0.333333.., 0.125125125..


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Example

Write in the form a/b (where a and b areintegers).

Soln:Define an equation

Let x=0.33333...(1) Multiply by 10 on both sides

10x=3.33333...(2)

(2)(1)

9x=3 Simplify

x=3/9=1/3

3.0


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Example 3: Converting decimal

numbers to fractions

Write as a rational fraction.

234.5


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Rational between 2 rational numbers

To find a rational number between 2 given

positive numbers is to find the average

between the 2 numbers

Eg; find a rational number between

is a rational between

4

1

3

1and

24

7

2

4

1

3

1

24

7

4

1

3

1and


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McKays Theorem

Finding a rational number between two given

positive rational numbers just by adding up

the numerator and the denominator,

Eg: find a rational number between

is between .

8

1

5

1and

13

2

85

11

13

2

8

1

5

1and

m15 rational and irrational numbers

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