Numbers and Operations

Families of numbers

Complex numbers

Real Numbers Imaginary Numbers

|Rational Numbers Irrational Numbers

|Integers

|Whole Numbers

|Natural Numbers

The Numbrella

Can be expressed as a fraction Can’t be expressed as a fraction

All “non-decimal” values

All positive integers and zero

All positive integers

i—or bi

a+biHas a real and an imaginary component

Counting Numbers◦ 1, 2, 3, 4, 5, …

Natural Numbers

Counting Numbers & Zero◦ 0, 1, 2, 3, 4, 5, …

Whole Numbers

Positive and Negative Numbers and Zero◦ …, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

Integers

Can be expressed as the ratio of 2 integers

Rational Numbers

n

m

Cannot be expressed as the ratio of 2 integers◦ Non-terminating, non-repeating integers◦ Π

Irrational Numbers

The approximate value of √7:√4 = 2 √9 = 3 so √7 is approx.

2.6

Determine the approximate value of the point:

1 2 3 4 5 6 7 8

The point is about 3.4

Examples:

Scientific Notation

1-9 are significant 0’s between digits are significant 0’s at the end suggest rounding and are

not significant Leading 0’s are not significant 0’s at the end of a decimal indicate the

level of precision Every digit in scientific notation is

significant

Significant Digits Rules

1024 4 Significant Digits

1000 1 Significant Digit

.0005 1 Significant Digit

ALWAYS HAVE ONE SIGNIFICANT DIGIT IN FRONT OF THE DECIMAL FOR SCIENTIFIC NOTATION

Examples

Expand: 2.15 x 10-3 2.15 x 103

a negative exponent tells you to move the decimal to the left

.00215 2150

Write in scientific notation: 3,145,062 2,230,000 .000345

move the decimal so that there is only one digit in front and count the number of spaces you have moved—moving left is positive here and right is negative.

3.145062 x 106 2.23 x 106 3.45 x 10-4

Examples

Simplify: do the math on the numeric portion as you normally would, use the rules of exponents on the powers of ten, place in standard scientific notation to finish (one digit before the decimal)

(2.75 x 102)(4 x 103)11 x 105

1.1 x 106

Examples

5 x 106 . 10 x 108

.5 x 10-2

5 x 10-1

Percent

Convert 20% to a decimal 20/100= .2

Convert .45 to a percentage .45 * 100= 45%

Convert ¾ to a percentage ¾= .75 .75 * 100=75%

Percentages

What is 7 percent of 50?◦ .07 * 50 = 3.5

A CD that normally costs $15 is on sale for 20% off. What will you pay◦ Option 1

.2 * 15 = 3 15-3= 12◦ Option 2

If it is 20% off you will pay 80% .8 * 15 = 12

Examples:

Order of Operations

PEMDASARAN THESIS

XPONENTS

MULT

&

DIV

ADD

&

SUB

From left to right

30 ÷ 10 • (20 – 15)2

30 ÷ 10 • 52

30 ÷ 10 • 25

75

Examples:

Parenthesis Exponents

then mult and divFrom left to right

Absolute Value

Formal definition

0 when x-

0 when x ||

x

xx

Absolute value is the distance from the origin and distance is always positive.

|6| |-7| |-9-3| 6 7 |-12|

12

Examples

GCF and LCM

GCF—greatest common factor What is the largest number that divides all the given

numbers evenly20 35 60 24

5 4 5 7 6 10 3 8

2 2 2 3 2 5 2 4

2 2

22* 5 5*7 22*3*5 23*3WHAT DO THEY SHARE?

5 22* 3=12

Examples

LCM—least common multiple What is the smallest number that the given number go

into evenly20 35 60 24

5 4 5 7 6 10 3 8

2 2 2 3 2 5 2 4

2 2

22* 5 5*7 22*3*5 23*3WHAT IS THE LAGEST VALUE SHOWN IN EACH?

22*5*7=140 23*3*5=120

Examples

Using Proportions

If Sue charges a flat rate each hour to babysit. If she ears $44 for 8 hours. What will she earn for 5 hours?

PRIMARY RULE:◦ If you put the $ amount in the numerator on one

side put the same value in the numerator on the other side. Etc.

cross mult. 220 = 8x27.5= x

Sue will earn $27.50 for 5 hours.

What is a proportion and how can you solve a problem with it?

58

44 x

Distance and Work Problems

Distance problems

rtD

TimeRateDistance

Example It took the Smith’s 5 hours to go 275 miles.

What was their average rate of speed?

D=rt275 = r(5)55 = r

They went about 55 mph

Use the reciprocal of the time for the rate of work

W for 1st

person =hours worked * rate of workW for 2

nd person =hours worked * rate of work

Total job always =1

1 = W for 1st

person + W for 2nd person

Work problems

John and Sam decide to build a bird house. John and build the bird house in 5 hours working alone. Sam can do it in 8 hours alone. How long will it take if they work together?

It will take them 3.08 hours to make the bird house.

Example:

851

xx

xx 5840 x1340 x08.3

EstimationWhat are the critical terms for estimation?

The “detail” associated with a measurement

Precision

Calculations with two different levels of precision can only be accurate to the least precise measure.

How correct a measurement is

The smaller the unit of measure the more accurate your measurement

Accuracy

The amount of difference between your measurement and the true value

Error

Jim bought 3 pounds of nails for $16.25. Which amount is closest to the price per pound?

Round off and check above and below15/3 = 5 and 18/3 = 6

A reasonable values would be between $5 and $6 but closer to $5

Examples:

Conversions

1 inch = 2.54 cm

12 inches = 1 foot

3 feet = 1 yard

5280 feet = 1 mile

How many inches are in 1 yard?◦ 1 yard = 3 feet 1 foot = 12 inches

3x12 =36 inches

Length Conversions

3 Teaspoons = 1 Tablespoon 2 Tablespoons = 1 ounce 8 ounces = 1 cup 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon

Fluid Conversions

16 ounces = 1 pound

2.2 pounds = 1 kilogram

2000 pounds = 1 ton

Weight Conversions

milli- centi-

-meter = distance -gram = weight -liter = fluid

kilo-