Welcome to our first seminar!

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The Real number system: Definitions

• Set: A collection of numbers (elements) represented by enclosing them in brackets

Example: { 1, 3 , 9 , 29}

• Natural numbers: counting numbers {1,2,3……….}

• Prime numbers: numbers that are divisible by themselves and 1 only.

Examples: 3,5,7,11,13,17,23 etc

• Composite numbers: Natural numbers that are not prime

• Whole numbers: Natural numbers plus zero, {0,1,2,3….}

• Integers: All whole negative and positive numbers and zero, {…. -3,-2,-1,0,1,2,3…..}

More definitions

• Rational numbers: numbers that can be represented by a/b and a and b are integers, b≠0;

Example 0/2 = 0, 3/7

• Irrational numbers: decimals that do not terminate or repeat.

Example π = 3.141592654……

These can be approximated using a calculator.

• Real numbers: Any number that can be represented on a number line, includes all rational and irrational numbers.

Prime numbers

Definition: A prime number is any whole number greater than 2 that can only be divided by itself and 1.

Other numbers have other factors. I like the factor tree method of factoring. The goal is to keep dividing the number by other numbers until each of them is prime at the bottom. I’ll do an example on the next page.

Example: find the factors of 108

108

2 54

2 27

3 9

3 3

2 3

108 = 2 2 3 3 3

108 = 2 3

The greatest common divisor: the largest natural number that divides into each member of a set.To find the GCF:l. Find the prime factorization of each member of the set.2. List all of the factors giving each the smallest exponent.3. Multiply all of those numbers in the list.

ExampleFind the GCF for 96 and 108

5

3 2

2

96 = 48 2 =24 2 2 = 12 2 2 2

= 6 2 2 2 2 = 3 2 2 2 2 2 2

96 3 2

108 = 54 2 = 27 2 2 = 9 3 2 2 3 3 3 2 2

108 = 3 2

GCF = 3 2 = 3(4) = 12 (smallest exponents)

The least common multiple (or denominator: the smallest number that can be divided into each member of a set (often fraction denominators)To find the LCM (LCD)1. Find the prime factorization of numbers in the set.2. List all factors giving the largest exponent.3. Multiply that list of numbers.

ExampleFind the LCD of 96 and 108

5

3 2

3 5

96 = 3 2

108 = 3 2

LCD = 3 2 27 32 864

Check by dividing this by both numbers:

864 96 9

864 108 8

Signed numbers

The easiest way to visualize a signed number is to look at a number line:

Think of it like this. If you have a – sign go left and a + sign go right

Subtracting Adding

Anything to the left is LESS than things to the right

For example: -6 is less than – 3 because it is to the left of it.

Rules about signed numbers

1. When adding numbers of the same sign, add and give them that sign.

Example: 2+4 = +6 -2 + (-4) = -6

To add positive numbers move right to 2 and then 4 more places to 6.

To add negative numbers move left to -2 and then 4 more places to -6.

2. To subtract numbers, subtract the smaller one from the larger one and give the answer the sign of the larger number.

Example: 2 – 5 Subtract:5 – 2 = 3 5 is larger and negative so

the answer is negative:2 – 5 = -3

On the number line go right 2 places to +2 and then left 5 places to -3

3. When multiplying or dividing signed numbers if there are an even number of negatives the answer is positive and if there are an odd number of negatives the answer is negative.

Examples:3(-5) 1 number (odd) is negative: the answer

is negative:= -15

-2(-9) 2 negatives (even): the answer is positive=18

(2)(-4)(-2)(-1) 3 negatives (odd); the answer is negative:

=-16 NOTE the importance of parenthesis

150 (-25) 1 negative number (odd): the answer is negative

= - 6

-3(-20) 3 negative numbers (odd): the answer is negative

-560

=-512

( 3) - means -1:

31( 3)

2 negatives (even): the answer is positive

33

= 3

1

3

3

(-5) The 3 means an odd number

of negatives, the answer is negative:

(-5)(-5)(-5)

=25(-5)

= - 125

-5 Since - 5 is NOT enclosed in

3

parenthesis, it may be written as:

-1(5)

=-1(5)(5)(5)

=-25(5)

=-125

Orders of operations

1. Parenthesis (innermost out)2. powers and roots3. multiply and divide 4. add and subtraction

Some people call this PEMDAS(Please excuse my dear aunt sally)

Or in a humorous alternative (Please embalm my dead Aunt Sally)

Parenthesis,Exponents,Multiply,Divide,Add,Subtract

A few more:

-8 + 3(-4-1) Use the orders of operation and do the ( ) first

= -8 + 3(-5)

= -8 + (-15)

= -23

-4(-3-1) - 5(-2-5) Use the orders of operation and do the ( ) first

= -4(-4) - 5(-2-5)

= -4(-4) - 5(-7)

= +16 - 5(-7)

= 16 - (-35)

= 16 + 35

= 51

Last ones of this type:

-5(-10) Do the top first:

-2+50

= 1 (odd) negative means that the answer is negative:-2

- 25

-3(-5) - (-2+3) Do the top first u

15-8

sing the orders of operation: do ( ) first:

-3(-5) - (+1)=

15-815 - 1

= 15-814

15-814

= 7

2

Fractions

Book definition: A fraction is any number that can be put in the form a/b where a and b are numbers and b is not equal to zero.

the numerator (top term) the denominator (bottom term)

We’ll talk about what a fraction means and some of the ways to work with them.

a

b

Properties

1. To find equivalent fractions (those that represent the same thing) you can use:If a, b, and c are numbers and b and c are NOT zero then:

For example:

a a c ac = =

b b c bc

1 1 4 4 = =

3 3 4 12

Reducing a fraction to lowest terms:1. Find the GCM2. Divide the top and bottom by the GCM

Example: 96 GCM = 12

10896 /12

= 108 /128

9

Changing improper fractions to mixed numbers

1. Divide the numerator by the denominator2. Write the quotient as the whole number part and the

remainder as the new numerator.

23 Divide the numerator (23) by the denominator (5)

5 4

5 23 5 goes into 23 4 times (4 is the whole number)

20 5(4) = 20; subtract 23-20=3

3 The remainer is 3 (this is the new numerator)

The whole number part is 5, the numerator is 3

and the denominator is the same; 4

3Answer 4

5

To change a decimal to a fraction

1. Write out the name.2. Put the original numbers on the top of the fraction.3. Put the place name on the bottom of the fraction

Example: Decimal: 0.54

Name: fifty four hundredths

54 Fraction:

10027

reduce to: 50

To add or subtract fractions:1. Find the LCD2. Form equivalent fractions.3. Add /subtract

Example:

7

96

7 5 - The LCD = 864;

96 108 9(96)=864 and 8(108)=864

9 7 8 5= -

9 96 8 108

63 40 - 864 86463 40

864

23 864

Irrational numbers: a number which is a non-terminating, non-repeating decimal. These are

radicals that have no root.

Rules:

1. Multiplying: multiply what is inside the .

For example: 7 5 35

2. Adding or subtracting: Treat them as "like" terms:

For example: 2 7 + 3 7 - 7

= (2+3-1) 7

= 4 7

But you cannot combine: 2 7 + 3 5

To simplify as square root:1. Write as any squares2. Take the things that are squared out as itself

2

96

16 6

4 6

4 6

The distributive property:a(b + c) = ab + ac

Examples:

3(2x - 3)

= 3(2x) + 3(-3)

= 6x - 9

2 5 - 2 3

2 5 + 2 2 3

5 2 - 2 6

Rules of exponents:

3 7

3 7 3+7 10

1. The bases must be identical.

For example x has the same base as x

2. When multiplying bases; add the exponents.

(x )(x ) = x = x

3. When dividing bases; subtract the e3

3-7 -47 4

3 7 3(7) 21

xponents.

x 1 = x = x or

x x4. When raising to a second exponent multiply

the exponents. (x ) = x = x

Examples

3 5 3 5 8

55 3 2

3

53 3(5) 15

2 2 = 2 = 2

2 = 2 = 2

2

2 = 2 = 2

Zero exponents

Anything to the zero power is ‘1’.

0

0

0

0

0

9 = 1

-9 the - is NOT included in the power so:

= -(9 ) 1

(-9) the - IS included in the power so:

(-9) 1 (because anything to the zero power is 1)

Negative exponentsA negative exponent means that you move the stuff raised to

that exponent to the other side of the fraction bar. For example:

-4-4

-44

2

2

22 2(2) 4

2

xx is actually Flip and change the sign of -4 to +4:

11

xx

And

3 Flip and change the -2 to +2:

x

x x x

3 3 9

Example-4

-4

4

- 5 Note that the - is not included in the power

5 This is actually -

1 so flip and change -4 to+4

1 = -

51

= - (5)(5)(5)(5)

1= -

625

Thank you for attending!