Real NumbersReal NumbersRational and IrrationalRational and Irrational

Here are the rational numbers represented on a number line.

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.

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.

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.

Integer Operations

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

a = (Vf - Vi) t

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.