Categories of Numbers

Real Numbers Irrationals Rationals Integers Whole Numbers Natural Numbers

If a number falls into a category, it automatically falls into all the categories above that category

There is no number which is both an irrational number and a rational number.

Natural Numbers – Counting numbers1, 2, 3, 4, 5, and so on. No Zero, no negatives, no decimals.1, 2, 3, 4, 5, 6, ...

Whole Numbers - Natural Numbers in it plus the number 0.0, 1, 2, 3, 4, 5, 6, ...

Integers - Whole Numbers and their opposites, or, Positive Numbers, Negative Numbers and Zero...., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ...

Rational Numbers - Any number that can be expressed as a ratio of two integers. Note the ratio in rational. Based on how the decimals act. Decimals either do not exist, as in 5, (which is 5/1). Or the decimals terminate, as in 2.4, which is 24/10. Or the decimals repeat with a pattern, as in 2.333, which is 7/3.

Summary: The behavior of the decimals is just the result when you divide an integer by another integer.

Irrational Numbers - This is any number that cannot be express as an integer divided by an integer, These numbers have decimals that never terminate and never repeat with a pattern.

• Order of Operations (STEPS)– Step 1: Evaluate the expressions inside grouping symbols– Step 2: Multiply and/or divide in order from left to right– Step 3: Add and/or subtract in order from left to right

• Expressions versus Equations (Discuss parts of an expression)– Numerical expression: contains a combination of numbers and

operations– Numerical equation: numerical expression with an equal sign– Algebraic expression: numerical expression that contains at least

one variable– Algebraic equation: algebraic expression with an equal sign

• Properties (RULES) – (Algebra pg 16)– Commutative

4+5 = 5+4 4x5 = 5x4– Associative

(4+5)+3 = 4+(5+3) (4x5)x3 = 4x(5x3)– Identity

3+0 = 3 3x0 = 0– Distributive

5(3+a) = 5(3)+5(a) 5(3)+5(a) = 5(3+a)

• Squares and Square Roots– The square of a number is the product of a number and itself

4 x 4 = 42 = 16– A square root of a number is one of its two equal factors 16 = 4

Integer Rules

Addition Rules

1. When the signs are the same, add the numbers and keep the sign.

7 + 8 = 15 -7 - 8 = -15

2. When the signs are different, subtract the numbers and take the sign of the larger number.

-7 + 8 = 1 7 – 8 = - 1

Note: Use Addition and Subtraction Rules when you “combine like terms”

Subtraction Rules Inverse (opposite) of Addition Rules

1. When the signs are the same, subtract the numbers and take the sign of the larger number.

7 – (+ 8) = - 1 -7 – (- 8) = 1

2. When the signs are different, add the numbers and keep the sign.

7 – (- 8) = 15 -7 – (+ 8) = -15

Multiplication & Division Rules 1. Negative x Negative = Positive Negative ÷ Negative = Positive

- 6 x (-3) = 18 -6 ÷ (- 2) = 3 2. Positive x Positive = Positive Positive ÷ Positive = Positive

6 x 3 = 18 6 ÷ 2 = 3 3. Negative x Positive = Negative Negative ÷ Positive = Negative

- 6 x 3 = -18 -6 ÷ 2 = -3 4. Positive x Negative = Negative Positive ÷ Negative = Negative

6 x (-3) = 18 6 ÷ (-2) = 3

STEPS - Order of Operations

G – Groupings Expressions inside grouping symbols

E – Exponents Exponents and square roots

M – Multiplication Multiplication from left to right

D – Division Division from left to right

A – Addition Addition from left to right

S – Subtraction Subtraction from left to right

(3 x 2)2 x 2 ÷ 9 +3 – 1 =

G (6)2 x 2 ÷ 9 +3 – 1 = E 36 x 2 ÷ 9 +3 – 1 = M 72 ÷ 9 +3 – 1 =D 8 + 3 – 1 =A 11 – 1 =S 10

Key Concepts (continued)• Absolute Value

– The absolute value of a number is the distance the number is from zero on the number line

• Comparing and Ordering Integers– When two numbers are graphed on a number line, the number

to the left is always less than the number to the right ( -8 < 8)

• Graphing Points– On a coordinate plane, the horizontal number line is the x-axis

and the vertical number line is the y-axis. The origin is (0,0) and is the point where the number lines intersect. The x-axis and y-axis separate the plane into four quadrants.

– Function: Relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Y

X

Quadrant I

Quadrant III

Quadrant II

Quadrant IV

0,0

Function versus Relation

X F(x) = X2

Y

2 22 4

4 42 16

6 62 36

8 82 64

10 102 100

X F(x) = ?

Y

2 4

4 16

2 8

5 25

5 10

Input InputOutput Output

Function Relation

Domain DomainRange Range

Changing Improper Fractions to Mixed Numbers

• An improper fraction is a fraction that has a numerator larger than or equal to its denominator.

• A proper fraction is a fraction with the numerator smaller than the denominator.

• A mixed number consists of an integer followed by a proper fraction.

Note the following pattern for repeating decimals:0.22222222... = 2/90.54545454... = 54/990.298298298... = 298/999Division by 9's causes the repeating pattern.

Note the pattern if zeros proceed the repeating decimal:0.022222222... = 2/900.00054545454... = 54/990000.00298298298... = 298/99900Adding zero's to the denominator adds zero's before the repeating decimal.

To convert a decimal that begins with a non-repeating part to a fraction: 0.21456456...Write it as the sum of the non-repeating part and the repeating part:  0.21 + 0.00456456456456456...

Next, convert each of these decimals to fractions. The first decimal has a divisor of power ten. The second decimal (which repeats) is converted according to the pattern given above.  21/100 + 456/99900

Next, add these fraction by expressing both with a common divisor and add:  20979/99900 + 456/99900 = 21435/99900

Finally, simplify it to lowest terms and check on your calculator or with long division:  1429/6660 = 0.2145645645…

Reflective Property:

Any quantity equal to itself (a = a)

Symmetric Property:

If one quantity equals a second quantity, then the second quantity equals the first

(a = b, then b = a)

Transitive Property:

If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity

(a = b, and b = c, then a = c)

Substitution Property:

A quantity may be substituted for its equal value

(a = b, then b can be substituted for a)

Problem: 2z - 5(z + 1) = 3z + 1

2z - 5z - 5 = 3z + 1

-3z - 5 = 3z + 1

-3z - 5 = 3z + 1 -3z -3z -6z - 5 = 0 + 1 +5 +5 -6z - 0 = 6

-6z = 6 ÷ -6 ÷ -6 z = -1

Remove all parentheses using the distributive property

Combine like terms that occur on the same

side of the equation using the commutative

and associative properties

Perform all addition and subtraction

operations

Perform all multiplication and division problems

Answer: z = -1

Greatest Common Factor

• The greatest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. There are three ways to find the greatest common factor.

• First method – Factorization TreeExample: Find the Prime Factors for 36 and 54.

36 54 6 6 6 9

2 3 2 3 2 3 3 3

– The prime factorization of 36 is 2 x 2 x 3 x 3– The prime factorization of 54 is 2 x 3 x 3 x 3– The prime factorizations of 36 and 54 both have one 2

and two 3s in common. Next, multiply these common prime factors to find the greatest common factor. Like this... 2 x 3 x 3 = 18

Third method - list all of the factors of each number, then list the common factors and choose the largest one.

Example: Find the GCF of 36 and 54.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.

The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18

Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

Second method – Division by a prime factor, then list the common factors and multiply them.

Example: Find the GCF of 36 and 54.

2 36 54 3 18 27 3 6 9 2 3The prime factorizations of 36 and 54 both have one 2 and two 3s in common. Next, multiply these common prime factors to find the greatest common factor. Like this... 2 x 3 x 3 = 18

Scientific Notation Scientific Notation is a special way of writing numbers that makes it easier to use big and small numbers.

a x 10n, where 1 a 10

Example: 102 = 100, so 700 = 7 × 102

7 × 102 is "Scientific Notation"

Example: 4,900,000,000

1,000,000,000 = 109 , so 4,900,000,000 = 4.9 × 109 in "Scientific Notation"

The number is written in two parts:

Just the digits (with the decimal point placed after the first digit), followed by × 10 to a power that puts the decimal point where it should be (i.e. it shows how many places to move

the decimal point).

In this example, 5326.6 is written as 5.3266 × 103, because 5326.6 = 5.3266 × 1000 = 5326.6 × 103

How to Do it

To figure out the power of 10, think "how many places do I move the decimal point?"

If the number is 10 or greater, the decimal point has to move to the left, and the power of 10 will be positive.

If the number is smaller than 1, the decimal point has to move to the right, so the power of 10 will be negative:

Example: 0.0055 would be written as 5.5 × 10-3

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3

Example: 3.2 would be written as 3.2 × 100

We didn't have to move the decimal point at all, so the power is 100

But it is now in Scientific Notation

Check

After putting the number in Scientific Notation, just check that:

The "digits" part is between 1 and 10 (it can be 1, but never 10) The "power" part shows exactly how many places to move the decimal point

Check

After putting the number in Scientific Notation, just check that:

The "digits" part is between 1 and 10 (it can be 1, but never 10) The "power" part shows exactly how many places to move the decimal point

Why Use It?

Because it makes it easier when you are dealing with very big or very small numbers, which are common in Scientific and Engineering work.

Example: it is easier to write (and read) 1.3 × 10-9 than 0.0000000013

It can also make calculations easier, as in this example:

Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.

What is its volume?

Let's first convert the three lengths into scientific notation:

width: 0.000 002 56m = 2.56×10-6 length: 0.000 000 14m = 1.4×10-7 height: 0.000 275m = 2.75×10-4

Then multiply the digits together (ignoring the ×10s):

2.56 × 1.4 × 2.75 = 9.856

Last, multiply the ×10s:

10-6 × 10-7 × 10-4 = 10-17 (this was easy: I just added -6, -4 and -7 together)

The result is 9.856×10-17 m3

It is used a lot in Science:

Example: Suns, Moons and Planets

The Sun has a Mass of 1.988 × 1030 kg.

It would be too hard for scientists to have to write 1,988,000,000,000,000,000,000,000,000,000 kg

Equation of a Straight Line The equation of a straight line is usually written this way:

y = mx + b

(or "y = mx + c" in the UK see below)

What does it stand for?

Slope (or Gradient) Y Intercept

y = how far up

x = how far along

m = Slope or Gradient (how steep the line is)

b = the Y Intercept (where the line crosses the Y axis)

How do you find "m" and "b"?

b is easy: just see where the line crosses the Y axis. m (the Slope) needs some calculation:

m =

Change in Y

Change in X

Knowing this we can work out the equation of a straight line:

Example 1

m = 2

1

= 2

b = 1 (where the line crosses the Y-Axis)

Therefore y = 2x + 1

With that equation you can now ...

... choose any value for x and find the matching value for y

For example, when x is 1:

y = 2×1 + 1 = 3

Check for yourself that x=1 and y=3 is actually on the line.

Or we could choose another value for x, such as 7:

y = 2×7 + 1 = 15

And so when x=7 you will have y=15

Example 2

m = 3

-1

= –3

b = 0

This gives us y = –3x + 0 We do not need the zero!

Therefore y = –3x

Example 3: Vertical Line

What is the equation for a vertical line? The slope is undefined ... and where does it cross the Y-Axis?

In fact, this is a special case, and you use a different equation, not "y=...", but instead you use "x=...".

Like this:

x = 1.5

Every point on the line has x coordinate 1.5, that’s why its equation is x = 1.5

Rise and Run

Sometimes the words "rise" and "run" are used.

Rise is how far up Run is how far along

And so the slope "m" is:

m = rise

run

You might find that easier to remember

Now Play With The Graph !

You can see the effect of different values of m (the slope) and b (the y intercept) at Explore the Straight Line Graph

Other Forms

We have been looking at the "slope-intercept" form. The equation of a straight line can be written in many other ways.

Another popular form is the Point-Slope Equation of a Straight Line.

Point-Slope Equation of a Line

The "point-slope" form of the equation of a straight line is:

y - y1 = m(x - x1)

Using this formula, If you know:

one point on the line and the slope of the line,

you can find other points on the line.

What does it stand for?

(x1, y1) is a known point

m is the slope of the line

(x, y) is any other point on the line

Making Sense of It

It is based on the slope:

Slope m = change in y

= y - y1

change in x x - x1

So this is the slope:

and we can rearrange it like this:

to get this:

So, it is just the slope formula in a different way!

Now let us see how to use it.

Example 1

slope "m" = 3

1

= 3

y - y1 = m(x - x1)

We know m now, and also know that (x1, y1) = (3,2), and so we have:

y - 2 = 3(x - 3)

That is a perfectly good answer, but we can simplify it a little:

y - 2 = 3x - 9

y = 3x - 9 + 2

y = 3x - 7

Example 2

slope "m" = 3

-1

= -3

y - y1 = m(x - x1)

We can pick any point for (x1, y1), so let's choose (0,0), and so we have:

y - 0 = -3(x - 0)

Which can be simplified to:

y = -3x

Example 3: Vertical Line

What is the equation for a vertical line? The slope is undefined!

In fact, this is a special case, and you use a different equation, like this:

x = 1.5

Every point on the line has x coordinate 1.5, that’s why its equation is x = 1.5

What About y = mx + b ?

You may already be familiar with the "y=mx+b" form.

It is the same equation, in a different form!

The "b" value (called the y-intercept) is where the line crosses the y-axis.

So point (x1, y1) is actually at (0, b)

and the equation becomes:

Start with y - y1 = m(x - x1)

(x1, y1) is actually (0, b): y - b = m(x - 0)

Which is: y - b = mx

Put b on other side: y = mx + b

And that is called the "slope-intercept" form of the equation of a line.

Absolute Value in Algebra Absolute Value means ...

... only how far a number is from zero:

"6" is 6 away from zero, and "-6" is also 6 away from zero.

So the absolute value of 6 is 6, and the absolute value of -6 is also 6

Absolute Value Symbol

To show you want the absolute value of something, you put "|" marks either side (called "bars"),

|-5| = 5 |7| = 7

More Formal

So, when a number is positive or zero we leave it alone, when it is negative we change it to positive.

This says: the absolute value of x equals:

x when x is greater than zero 0 when x equals 0

-x when x is less than zero (this "flips" the number back to positive)

Example: what is |-17| ?

Well, it is less than zero, so we need to calculate "-x":

- ( -17 ) = 17

(Because two minuses make a plus)

Useful Properties

Here are some properties of absolute values that can be useful:

|a| ≥ 0 always!

That makes sense ... |a| can never be less than zero.

|a| = √(a2)

Squaring a makes it positive or zero (for a as a Real Number). Then taking the square root will "undo" the squaring, but leave it positive or zero.

|a × b| = |a| × |b|

Means these are the same:

the absolute value of (a times b), and (the absolute value of a) times (the absolute value of b).

Which can also be useful when solving

|u| = a is the same as u = ±a and vice versa

Which is often the key to solving most absolute value questions.

Example: solve |x+2|=5

Using "|u| = a is the same as u = ±a":

this: |x+2|=5 is the same as this: x+2 = ±5

Which will have two solutions:

x+2 = -5 x+2 = +5

x = -7 x = 3

Graphically

Let us graph that example:

|x+2| = 5

It is easier to graph if you have an "=0" equation, so subtract 5 from both sides:

|x+2| - 5 = 0

And here is the plot of |x+2|-5, but just for fun let's make the graph by shifting it around:

Start with |x| then shift it left to make it |x+2| then shift it down to make it |x+2|-5

And you can see the two solutions: -7 or +3.

Absolute Value Inequalities

Mixing Absolute Values and Inequalites needs a little care!

There are 4 inequalities:

< ≤

> ≥

less than less than

or equal to greater than

greater than or equal to

Less Than, Less Than or Equal To

With "<" and "≤" you get one interval centered on zero:

Example: Solve |x| < 3

This means the distance from x to zero must be less than 3:

Everything in between (but not including) -3 and 3

It can be rewritten as:

-3 < x < 3

And as an interval it can be written as: (-3, 3)

The same thing works for "Less Than or Equal To":

Example: Solve |x| ≤ 3

Everything in between and including -3 and 3

It can be rewritten as:

-3 ≤ x ≤ 3

And as an interval it can be written as: [-3, 3]

How about a bigger example?

Example: Solve |3x-6| ≤ 12

Rewrite it as:

-12 ≤ 3x-6 ≤ 12

Add 6:

-6 ≤ 3x ≤ 18

Lastly, multiply by (1/3). Because you are multiplying by a positive number, the inequalities will not change:

-2 ≤ x ≤ 6

Greater Than, Greater Than or Equal To

This is different ... you get two separate intervals:

Example: Solve |x| > 3

It looks like this:

Up to -3 or from 3 onwards

It can be rewritten as

x < -3 or x > 3

As an interval it can be written as: (-∞, -3) U (3, +∞)

Careful! Do not write it as

-3 > x > 3

"x" cannot be less than -3 and greater than 3 at the same time

It is really:

x < -3 or x > 3

"x" is less than -3 or greater than 3

The same thing works for "Greater Than or Equal To":

Example: Solve |x| ≥ 3

Can be rewritten as

x ≤ -3 or x ≥ 3

As an interval it can be written as: (-∞, -3] U [3, +∞)

To see what the graph of y = |x| looks like, let’s create a table of values.

x y

-3 3

-2 2

-1 1

0 0

1 1

2 2

3 3

To graph these values, simply plot the points and see what happens.

Whenever you have an absolute value graph, the general shape will look like a “v” (or in some cases, an upside down “v” as we will see later). Let's Practice:

i. Graph y = |x+2|

We know what the general shape should look like, but let’s create a table of values to show exactly how this graph will look.

x y

-3 |-3 + 2| = |-1| = 1

-2 |-2 + 2| = |0| = 0

-1 |-1 + 2| = |1| = 1

0 |0 + 2| = |2| = 2

1 |1 + 2| = |3| = 3

2 |2 + 2| = |4| = 4

3 |3 + 2| = |5| = 5

So our graph of y = |x + 2| looks like

Graphing Absolute Value Functions

i. Graph y = |x| - 4

The table of values looks like this:

x y

-5 5 - 4 = 1

-4 4 - 4 = 0

-3 3 - 4 = -1

-2 2 - 4 = -2

-1 1 - 4 = -3

0 0 - 4 = -4

1 1 - 4 = -3

Which makes the graph look like this:

Notice that the graph in this example is the same shape as except that it has been moved down 4 units.

ii. Graph y = -|x|

In creating the table of values, be careful of your order of operations. You should find the absolute value of x first and then change the sign of that answer.

x |x| y

-2 2 -2

-1 1 -1

0 0 0

1 1 -1

2 2 -2

So the graph of looks like:

In this example, we have the exact same shape as the graph of y = |x| only the “v” shape is upside down now.

Some things to Remember when converting any type of measures: To convert from a larger to smaller metric unit you always multiply To convert from a smaller to larger unit you always divide

The latin prefixes used in the metric system literally mean the number they represent. Example: 1 kilogram = 1000 grams. A kilo is 1000 of something just like a dozen is 12 of something.

This is the metric conversion stair chart. You basically take a place value chart turn it sideways and expand it so it looks like stairs. The Latin prefixes literally mean the number indicated. Meter, liter or gram can be used interchangeably.

You use this chart to convert metric measurements like this: • If you are measuring length use meter. • If you are measuring dry weight use grams.• If you are measuring liquid capacity use liter

For every step upward on the chart you are dividing by 10 or moving the decimal one place to the left. Example: To convert 1000 milligrams to grams you are moving upward on the stairs. Pretend you are standing on the milli-gram stair tread and to get to the 1-gram stair tread you move up 3 steps dividing by 10 each time.

1000/10 = 100 100/10 = 10 10/10 = 1 or 1000/1000 = 1

or use the shortcut and just move the decimal place one place to the left with each step 1000 milligrams = 1 gram. When you move down the stairs you are multiplying by 10 for each step. SO you are adding a zero to your original number and moving the decimal one place to the right with each step. Example: To convert 2 kilometers to meters you move 3 steps down on the chart so you add 3 zeros to the 2.

2 kilometers = 2000 meters Problems: 1.) 3 meters = _300_ centimeters (multiply by 100 or just add 2 zeros)2.) 40 liters = _4_ dekaliters (dividing by 10, or move your decimal place one place to the left)3.) 600 milligrams = _0.6_ grams (dividing by 1000, or move your decimal 3 places to the left)4.) 5 kilometers = _50_ hectometers 5.) 70 centimeters = _0.7_ meters 6.) 900 deciliters= _9_ dekaliters 7.) John's pet python measured 600 centimeters long. How many meters long was the snake? 6m8.) Faith weighed 5 kilograms at birth. How many grams did she weigh? 500 grams9.) Jessica drank 4 liters of tea today. How many deciliters did she drink? 40 deciliters

Metric Measures and Conversions

Unit Equivalent Unit

1 foot __12___ inches

1 yard ___3___ feet

1 year ___7___ weeks

1 pound __16___ ounces

1 gallon ___4___ quarts

1 quart ___2___ pints

COMMON MEASUREMENTS