REAL NUMBERS Rational and Irrational

Real NumbersReal NumbersRational and IrrationalRational and Irrational

Let’s look at the relationships between number sets. Notice rational and

irrational numbers make up the larger number set known as Real Numbers

A number represents the value or quantity of something… Like how much money you have.. Or how many marbles

you have… Or how tall you are.As you may remember from earlier grades

there are different types of numbers.

A number line - is an infinitely long line whose points match up with the real

number system.

Here are the rational numbers represented on a number line.

IntegersIntegersThe coldest temperature on record in The coldest temperature on record in the U.S. is -80° F, recorded in 1971 in the U.S. is -80° F, recorded in 1971 in

AlaskaAlaska

Integers are used to represent real-world quantities such as temperatures, miles per hour, making withdrawals from your bank account, and other quantities. When you

know how to perform operations with integers, you can solve equations and

problems involving integers.

By using integers, you can express elevations above, below, and at sea level.

Sea level has an elevation of 0 feet. Badwater Basin in Utah is -282 below sea level, and Clingman’s Dome in the Great Smokey Mountains is +6,643 above sea

level.

If you remember, the whole numbers are the counting numbers and zero:

0, 1, 2, 3,…Integers - the set of all whole numbers and their opposites. This means all the positive integers and all the negative

integers together.

Opposites – two numbers that are equal distance from zero on a number line; also

called additive inverse.The additive inverse property states that if you add two opposites together their sum

is 0-3 + 3 = 0

Integers increase in value as you move to the right along a number line. They

decrease in value as you move to the left. Remember to order numbers we use the

symbol < means “less than,” and the symbol > means “is greater than.”

A number’s absolute value - is it’s distance from 0 on a number line. Since distance can

never be negative, absolute values are always positive. The symbol || represents the absolute value of a number. This symbol is

read as “the absolute value of.” For example |-3| = 3.

Finding absolute value using a number line is very simple. You just need to know the

distance the number is from zero. |5| = 5, |-6| = 6

Lesson Quiz

Compare, Use <, >, or =.1) -32 □ 322) 26 □ |-26|3) -8 □ -124) Graph the numbers -2,

3, -4, 5. and -1 on a number line. Then list the numbers in order from least to greatest.

5) The coldest temperature ever recorded east of the Mississippi is fifty-four degrees below zero in Danbury, Wisconsin, on January 24, 1922. Write the temperature as an integer.

Integer OperationsInteger OperationsRules for Integer OperationsRules for Integer Operations

Adding IntegersWhen we add numbers with the same signs,

1) add the absolute values, and2) write the sum (the answer) with the sign of the

numbers.When you add numbers with different signs,

1) subtract the absolute values, and2) write the difference (the answer) with the sign of the number having the larger absolute value.

Try the following problems

1) -9 + (-7) = -16

2) -20 + 15 = -5

3) (+3) + (+5) = +8

4) -9 + 6 = -3

5) (-21) + 21 = 0

6) (-23) + (-7) = -30

Subtracting IntegersYou subtract integers by adding its

opposite.9 – (-3)

9 + (+3) = +12

-7 – (-5)-7 + (+5) = -2

Try the following problems

1) -5 – 4 = -5 + (-4) = -9

2) 3 – (+5) =3 + (-5) = -2

3) -25 – (+25) =-25 + (-25) = -50

4) 9 – 3 =9 + (-3) = +6

5) -10 – (-15) =-10 + (+15) = +5

Multiplying and Dividing IntegersIf the signs are the same,

the answer is positive.

If the signs are different,the answer is negative.

Try the following problems

Think of multiplication as repeated addition.3 · 2 = 2 + 2 + 2 = 6 and 3 · (-2) = (-2) + (-2) + (-

2) = -6

1) 3 · (-3) = Remember multiplication is fast adding

= 3 · (-3) = (-3) + (-3) + (-3) = -9

2) -4 · 2 = Remember multiplication is fast adding

= -4 · 2 = (-4) + (-4) = -8

Dividing Integers

Multiplication and division are inverse operations. They “undo” each other. You can use this fact to discover the rules for division of integers.4 · (-2) = -8 -4 · (-2) = 8-8 ÷ (-2) = 4 8 ÷ (-2) = -4

same sign positive different signs negativeThe rule for division is like the rule for multiplication.

Try the following problems

1) 72 ÷ (-9) 72 ÷ (-9) Think: 72 ÷ 9 = 8

-8The signs are different, so the quotient is negative.

2) -144 ÷ 12 -144 ÷ 12 Think: 144 12 = 12

-12 The signs are different, so the quotient is negative.

3) -100 ÷ (-5) Think: 100 ÷ 5 = 20

-100 ÷ (-5) The signs are the same, so the quotient is

positive.

Lesson Quiz

Find the sum or difference1) -7 + (-6) =2) -15 + 24 + (-9) =Evaluate x + y for x = -

2 and y = -153) 3 – 9 =4) -3 – (-5) =Evaluate x – y + z for x = -4, y = 5, and z = -

10

Find the product or quotient1) -8 · 12 =2) -3 · 5 · (-2) =3) -75 ÷ 5 =4) -110 ÷ (-2) =5) The temperature in

Bar Harbor, Maine, was -3 F. During the night, it dropped to be four times as cold. What was the temperature then?

Rational NumbersRational NumbersFractions and DecimalsFractions and Decimals

Rational numbers – numbers that can be written in the form a/b (fractions), with

integers for numerators and denominators.

Integers and certain decimals are rational numbers because they can be written as

fractions. a 1 2 3 4 5 … b 1 1/ 1 2/ 1 3/ 1 4/ 1 5/ 1 … 2 1/ 2 2/ 2 3/ 2 4/ 2 5/ 2 … 3 1/ 3 2/ 3 3/ 3 4/ 3 5/ 3 … 4 1/ 4 2/ 4 3/ 4 4/ 4 5/ 4 … 5 1/ 5 2/ 5 3/ 3 4/ 5 5/ 5 … …

Remember you can simplify a fraction into a decimal by dividing the denominator into the numerator, or you can reduce a decimal by

placing the decimal equivalent over the appropriate place value.

O.625 = 625/1000 = 5/8

Hint: When given a rational number in decimal form (such as 2.3456) and asked to

write it as a fraction, it is often helpful to “say” the decimal out loud using the place

values to help form the fraction.2 . 3 4 5 6 o a t h t ten- n n e u h t e d n n o h s t d u o h r s u s e a s d n a T d n h t d s h t s h s

Write each rational number as a fraction:

Rational number I n decimal f orm

Rational number I n f ractional f orm

0.3 3/ 10 0.007 7/ 1000 -5.9 -59/ 10

Hint: When checking to see which fraction is larger, change the fractions to decimals by dividing and comparing their decimal

values.

Which of the given numbers is greater?

Using f ull calculator display to compare the numbers

2/ 3 and 1/ 4 .6666666667 > .25 -7/ 3 and – 11/ 3 -2.333333333 > -3.666666667

Examples of rational numbers are:

6 or 6/1 can also be written as 6.0-2 or -2/1 can also be written as -2.0½can also be written as 0.5-5/4 can also be written as -1.252/3 can also be written as .662/3 can also be written as 0.666666…21/55 can also be written as 0.38181818…53/83 can also be written as 0.62855421687…

the decimals will repeat after 41 digits

Examples: Write each rational number as a fraction:

1) 0.3

2) 0.007

3) -5.9

4) 0.45

Since Real Numbers are both rational and

irrational ordering them on a number line can be difficult if you don’t pay attention to the details.As you can see from the

example at the left, there are rational and irrational

numbers placed at the appropriate location on

the number line.This is called ordering

real numbers.

Irrational numbersIrrational numbers√√2 = 1.414213562…2 = 1.414213562…

no perfect squares hereno perfect squares here

Irrational number – a number that cannot be expressed as a ratio of two integers (fraction) or as a repeating or

terminating decimal.

• An irrational number cannot be expressed as a fraction.

• Irrational numbers cannot be represented as terminating or repeating decimals.

• Irrational numbers are non-terminating, non-repeating decimals.

Below are three irrational numbers.Decimal representations of each of these are

nonrepeating and nonterminating

10101001000.0

Examples of irrational numbers

are:

= 3.141592654…√2 =

1.414213562…0.12122122212 √7, √5, √3, √11,

343√Non-perfect squares

are irrational numbers

Note:The √ of perfect

squares are rational numbers.

√25 = 5 √16 = 4 √81 = 9Remember: Rational numbers when divided

will produce terminating or repeating decimals.

NOTE:Many students think that

is a terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use as 3.14, but that problem is rounding the value of to make your calculations easier. It is actually an infinite decimal and is an irrational number.

There are many numbers on a real

number line that are not rational. The number is not a

rational number, and it can be located on a real number line by

using geometry. The number is not equal to 22/7, which is only an approximation of

the value. The number is exactly equal to

the ratio of the circumference of a

circle to its diameter.

Enjoy your Pi