Ideas
Addition and subtraction of integers
When there are two operators, or mathematical signs written together, you can simplify them.
+(-) or -(+) becomes a negative (-)
-(-) becomes a positive (+)
The two negative signs have to be right next to each other.
-4-8 means "-4 minus -8". There is no change to the signs here.
-4-(-8) means "-4 minus -8". This becomes "-4 plus 8". In this case, the operators changed.
The same rules apply, even when there are more than 2 numbers.
Examples
Example 1
Evaluate: -7-(-2)-(-6).
The two negative signs have to be right next to each other.
-4-8 means "-4 minus -8". There is no change to the signs here.
-4-(-8) means "-4 minus -8". This becomes "-4 plus 8". In this case, the operators changed.
The same rules apply, even when there are more than 2 numbers.
Multiplication and division of integers
Product means multiplication.
Quotient means division.
The multiplying or dividing two number with the same sign gives a positive integer.
The multiplying or dividing two number with the different sign gives a negative integer.
Examples
Example 2
Evaluate: -3 \times 4 \times (-3).
When working out the sign, remember:
- \text{Positive}\times \text{Positive} = \text{Positive}
- \text{Negative}\times \text{Negative} = \text{Positive}
- \text{Positive}\times \text{Negative} = \text{Negative}
- \text{Negative}\times \text{Positive} = \text{Negative}
Square and cube numbers
Squaring a number means multiplying it by itself. As we have just seen that the rule for multiplying two numbers with the same sign results in a positive number, all square numbers will be positive.
For example:
squaring the number 4, means 4^{2}=4\times 4 = 16
squaring the number -5 means (-5)^{2}=(-5)\times (-5)=25
Cubing a number means multiplying it by itself and then multiply it by itself again. Here are some examples of cubing positive or negative numbers.
For example:
cubing the number 4 means 4^{3}=4\times 4 \times 4 = 64
cubing the number -5 means (-5)^{3}=(-5)\times (-5)\times (-5)=-125
Be careful:
-3^{2} is not the same as (-3)^{2}.
-3^{2} means -(3^{2}) or -1\times (3\times 3), which gives an answer of -9 because we are squaring 3 and then multiplying by -1.
(-3)^{2} means (-3)\times (-3), which gives an answer of 9 because the brackets mean we are squaring -3.
Examples
Example 3
Evaluate: 5^{3}-2^{2}+10.
Squaring a number means multiplying it by itself.
Cubing a number means multiplying it by itself and then multiply it by itself again.
Square and cube roots
Finding the square root of a value is the inverse (opposite) operation of squaring a value. This is represented as a square root symbol written with a number inside it, for example, \sqrt{25}. This means find the square root of 25.
Finding the cube root of a value is the inverse operation to cubing a value. This is represented using the cube root symbol written with a number inside it. For example, ^{3}\sqrt{125}. This means find the cube root of 125.
Examples
Example 4
Evaluate \sqrt{5^2+12^2}.
The square root of a value is the inverse (opposite) operation of squaring a value.
The cube root of a value is the inverse operation to cubing a value.
Order of operations
The order in which we do things is important. For example, put on socks then shoes, rather than shoes and then socks. The same goes for solving maths problems with more than one operation
There are rules to be followed in order to solve maths problems correctly. The order of operations is:
Step 1: Do operations inside brackets (...).
Step 2: Do multiplication and division going from left to right.
- Step 3: Do addition and subtraction going from left to right.
Examples
Example 5
Evaluate: ((36-(10+10))\div2) + 14 \times 6.
The order of operations is:
Step 1: Do operations inside brackets (...).
Step 2: Do multiplication and division going from left to right.
- Step 3: Do addition and subtraction going from left to right.