This is a draft cheat sheet. It is a work in progress and is not finished yet.
Integers
Integers are positive whole numbers, negative whole numbers and zero. |
When there is more than 1 operation, remember to use BODMAS. |
When adding/subtracting, look at the symbols in the middle. |
When multiplying/dividing, look at the symbols next to the numbers. |
+ + = + |
- - = + |
+ - = - |
- + = - |
Indices
The index is the small number above the base. |
Example: 24 |
2 is the base, 4 is the index. |
24 can also be written as 2 x 2 x 2 x 2. |
24 can also be written as 16, as 2 x 2 x 2 x 2 = 16. This is known as a basic numeral. |
Reciprocals
The reciprocal is simply: 1/number. |
Reciprocal: What to multiply a value by to get 1. It is also known as "Multiplicative Inverse". |
Example: The reciprocal of 2 is ½ (a half). |
More Examples: |
Number |
Reciprocal |
As a decimal |
5 |
1/5 |
= 0.2 |
8 |
1/8 |
= 0.125 |
1000 |
1/1000 |
= 0.001 |
For fractions, flip the whole fraction over |
Example: The reciprocal of 3/4 is 4/3 |
Every number has a reciprocal except 0. |
Multiplying a number by its reciprocal gets us 1. |
|
|
Simplifying Expressions
How to simplify an expression: |
1. Remove brackets by multiplying factors. |
2. Use index laws to remove brackets in terms with indices. |
3. Combine like terms by adding coefficients. |
4. Combine the constants. |
Variable: A symbol for a number we don't know yet. It is usually a letter like x or y. |
Constant: A number on its own. |
Coefficient: A number used to multiply a variable. |
Variables without a number have a coefficient of 1. |
Example: ax2 + bx + c |
x is a variable, a and b are coefficients and c is a constant. |
Like terms are terms whose variables (and their exponents such as the 2 in x2) are the same. In other words, terms that are "like" each other. (Note: the coefficients can be different) |
Example: |
−2xy2 |
6xy2 |
(1/3)xy2 |
These are all like terms because the variables are all xy2 |
Prime and Composite Numbers
A prime number is a number that can be divided evenly only by 1, or itself. And it must be a whole number greater than 1. |
A composite number is a whole number that can be divided evenly by numbers other than 1 or itself. |
Factors and Multiples
Factors and multiples are both to do with multiplication: |
Factors are what we can multiply to get the number. |
Multiples are what we get after multiplying the number by an integer (not a fraction). |
|
|
Index Laws
1. The numbers in index form with the same base can be multiplied together by being written in factor form first. |
Multiply: am x an = am + n |
2. The numbers in index form with the same base can be divided first by being written in factor form. |
Divide: am ÷ an = am - n |
3. Any base that has an index power of 0 is equal to 1. |
Zero Law: a0 = 1 |
4. Every number and variable inside the brackets should have its index multiplied by the power outside the brackets. |
Powers: (am)n = am x n |
5. Negative Indices: a-3 = 1 ÷ a3 |
6. Any number or variable that does not appear to have an index really has an index of one. |
7. Every number or variable inside the brackets must be raised to the power outside the brackets. |
Factor Trees
A factor tree is a special diagram where you find the factors of a number, then the factors of those numbers, etc until you can't factor any more. |
The ends are all the prime factors of the original number. |
A prime factor is a factor that is a prime number: one of the prime numbers that, when multiplied, give the original number. |
Example: The prime factors of 15 are 3 and 5 (3×5=15, and 3 and 5 are prime numbers). |
There is only one (unique) set of prime factors for any number. This is called the Fundamental Theorem of Arithmetic. |
|
