Numbers can be represented along a one dimensional line starting at zero we might have a line that looks like this:

0 1 2 3 4

On the number line, numbers which are to the left are less than numbers to the right. We use the symbol “<” to indicate that one number is smaller than another number, and we use the symbol “>” to represent that one number is larger than another number.

Example

3 > 2 is used to represent “three is greater than two”

5 < 9 is used to represent “five is less than nine”

We call the set of counting numbers whole numbers {0, 1, 2, 3, 4, 5, …}
Note that there is some disagreement about weather or not zero should be include din this set of numbers.

Not only do we have positive numbers we also have negative numbers. Negative numbers are one the challenges many people have in mathematics. A negative number is any number whose value is less than zero. To represent a negative number we place a minus sign in front of the number.

Example

- 2
- 4/5
- 6.28

The set of integers does not include decimals or fractions only whole numbers.

The distance from 0 to an integer along a number line is its absolute value. The absolute value of a number is written using vertical lines as brackets around the number.

Example

| 2 | = 2
| -6 | = 6

Example

2 + 3 means you start at the number 2 and move 3 in the positive direction
So our answer would be 5

This process is the same if we have a negative value as well.

Example

-3 + 6 means you start at -3 and count 6 steps in the positive direction
So your answer will be 3

To subtract integers you simply need to remember that subtracting an integer is the same as adding it’s opposite.

Example

5 – 6 = 5 + (-6) = -1
9 – 4 = 9 + (- 4) = 5

Or

Example

Let’s try an example. Lets say we had 6 groups of trees and each group had 3 trees meaning there are 18 trees altogether. So 3 x 6 = 18.

Remember multiplication is commutative, a x b = b x a. Any number times 0 is equal to 0, and any number times one is equal to that number.

Multiplication of negative numbers can be a little tricky. When we multiply a positive number by a negative number we get a negative result.

Example

3 x -7 = -21

Example

If we have 20 / 4 (twenty divided by 4) we could rewrite this as 4 times what = 20. When we solve this equation we see that 4 times 5 equals 20 so 20 dived by four equals 5.

Just like integer multiplication we have rules that govern integer division. Let’s start by thinking about a log that is 12 feet long, if we divide this log into 1 piece it will still be 12 feet long, so any number divided by 1 is that number. Now since a log with no length, that is length equals 0, will not exist it is impossible to divide by zero. So we say the result is undefined. Just like in multiplication when we have two integers with different signs we will get a negative integer, and when we have two integers with the same sign we get a positive integer.

Example

By closed we mean when we add, subtract, multiply or divide 2 integers we get an integer.

When we say commutative it means that no matter what order we add, subtract, multiply or divide 2 integers we get an integer.

Example

4/5 is rational since 4 and 5 are both integers
7 is rational number since it can be expressed as 7/1 and 7 and 1 are both integers
2.4 is rational since it can be expressed as 12/5 and both 12 and 5 are integers

Rational numbers either have a terminating decimal such as 6.254 or a repeating decimal such as 0.3333333

Irrational numbers on the other hand can not be expressed in the form x / y, and have a never ending and never repeating decimal. For example pie is an irrational number.

Other types of real numbers include prime numbers. Prime numbers are a positive integer which has exactly two factors, 1 and itself. The first 4 prime numbers are 2, 3, 5, and 7.

4 + 3 x 5

We could either add the 4 and 3 and than multiply the result by 5 giving us 35 or we could multiply the 3 and 5 and than add the 4 giving us giving us 19.

As you can see depending upon the order we complete the operations we get two very different answers, so which is correct?

Well to help reduce confusion about the order of operations you can use the following convention. BODMAS is the agreed convention for answering problems that are a mixture of adding, subtracting, multiplying and dividing. It stands for

B – do what is inside the brackets first
O – Order is important in math
DM – do division and multiplication as you come to them from left to right
AS - do addition and subtraction as you come to them from left to right.

So for the problem above the correct solution would be:

4 + 3 x 5
= 4 + 15
= 19

No let’s look at a few other examples:

Example 1

18 - 6 x 3
= 18 – 18
= 0

Example 2

(18 – 6) x 3
= (12) x 3
= 36

Example

For 6³ we say that 6 is the base and 3 is the power or exponent.
This means that we multiply 6 by itself 3 times.

6³ = 6 x 6 x 6 = 216

There are a few special cases of exponents that we should talk about, specifically when we have an exponent of 1, 0 or a negative number.

In general, any number x raised to a power of 1 is x. Where as any number x raised to the power of zero is one. When we have a negative number as the power the result is 1/x.

5⁴ x 5⁶ can be written as (5 x 5 x 5 x 5) x (5 x 5 x 5 x5 x 5 x 5)

Notice we have 4 fives in the first bracket and 6 fives in the second bracket, so altogether we will have 4 + 6 = 10 fives multiplied together. So we can rewrite the equation as the following:

5⁴ x 5⁶ = 5⁴⁺⁶ = 5¹⁰

Example

Now let’s go over an example of how to divide two exponents with the same base.

Let’s divide 4⁵ by 4²

So we have 4⁵ / 4² = (4 x 4 x 4 x 4 x4) / (4 x 4)

If we cancel out two top ours with the two bottom fours we get:

= 4³
= 6⁴

Example

As an example let’s use the following:

(5²)³ = 5² x 5² x 5²

Base don the rules we use din the multiplication examples earlier we can see that this will result in 56. We could have also done this by the following:

(5²)³= 5² x ³ = 56

Now let’s raise a product to a power. Here is our example:

(6 x 3)²

To solve this problem we raise each number inside the brackets to the power and than multiply them together.

(6 x 3)² = 6² x 3² = 324

When we are raising a quotient to a power we can solve it in a similar manner. Here is an example:

(4/5)³ = 4³/5³ = 0.512

Example

So for example 15^1/2 = √15 = 3.873

You can also have cubed roots, fourth roots and so on.

There are a few key rules to working with roots. First if a and b are both greater or equal to 0 than:

√ a x b = √a x √b

However this only applies when multiplying, if we are adding

√ a + b ≠ √a + √b

This is important because our calculators will often provide us an answer which has many decimal places in it and we need to know just how many we should use.

Significant digits provide an indication of the accuracy of a number. A digit which is 0 is significant only if it is not a place holder.

Example

Example 

If we compare two numbers lets say 0.057 and 3.563, we see that 3.563 is more accurate because it has four significant digits, where 0.041 only has two. We also notice that the two numbers have the same precision, since the last significant digit is in the thousandths position for both.

Example

2.3 + 6.528 + 13. 45 = 22.3
Notice that the answer in this example has the same precision of the least precise value.

When multiplying or dividing approximate numbers, the result should have the accuracy of the least accurate number.

Example

5.625 x 3.87 = 21.76875
But since we have only 3 significant digits the answer becomes ≈ 21.8

Lets say you had 550,000 trees you were managing, instead of writing this out you could convert this number to scientific notation.

Example

550,000 (ordinary notation)
= 5.5 x 100,000
= 5.5 x 10⁵