Here in performing calculations with numbers we concentrate on the operations, +, −, × and ÷.
We agree that 6+8 is the sum of 6 and 8. Note that 6+8 is equal to 8+6 so that the order in which we write down the numbers does not matter when we are adding them. Because the order does not matter, addition is said to be commutative. This ﬁrst property is called commutativity. When more than two numbers are to be added, as in 4 + 8 + 9, it makes no diﬀerence whether we add the 4 and 8 ﬁrst to get 12 + 9, or whether we add the 8 and 9 ﬁrst to get 4 + 17. Whichever way we work we will obtain the same result, 21. Addition is said to be associative. This second property is called associativity
We say that 8 − 3 is the diﬀerence of 8 and 3. Note that 8 − 3 is not the same as 3 − 8 and so the order in which we write down the numbers is important when we are subtracting them i.e. subtraction is not commutative. Subtracting a negative number is equivalent to adding a positive number, thus 9−(−3) = 9 + 3 = 12.
The plus or minus sign (±)
In mathematical analysis we often use the notation plus or minus, ±. For example, we write 14±6 as shorthand for the two numbers 14 + 6 and 12−6, that is 20 and 6. If we say a number lies in the range 14±6 we mean that the number can lie between 6 and 20 inclusive.
The instruction to multiply, or obtain the product of, the numbers 5 and 7 is written 5×7. Sometimes the multiplication sign is missed out altogether and we write (5)(7). Note that (5)(7) is the same as (7)(5) so
Also check: How to solve Quadratic Equations
multiplication of numbers is commutative. If we are multiplying three numbers, as in 3×4×5, we will get the same result whether we multiply the 3 and 4 ﬁrst to obtain 12×5, or whether we multiply the 4 and 5 ﬁrst to obtain 3×20. Either way the result is 60. Multiplication of numbers is associative. When multiplying numbers:
- positive × positive = positive
- negative × negative = positive
- positive × negative = negative
- negative × positive = negative
consider, (−6)×4 = −24, and (−5)×(−6) = 30. If you are working with fractions we sometimes use the word ‘of’ as in ‘ﬁnd 12 of 36’. In this context ‘of’
is equivalent to multiply, that is 10 of 36 is equivalent to 10 ×36 = 360
Division (÷) or (/)
The quantity 9 ÷ 3 means 9 divided by 3. This is also written as 9/3 or and is known as the quotient of 9 and 3. In the fraction the top line is called the numerator and the bottom line is called the denominator. Note that 8/4 is not the same as 4/8 and so the order in which we write down the numbers is important. Division is not commutative. When dividing numbers:
- Positive/positive = positive
- Positive/negative = negative
- Negative/positive = negative
- Negative/negative = positive
The reciprocal of a number
The reciprocal of a number is found by inverting it. If the number is inverted we get . So the reciprocal of is . Because we can write 7 as , the reciprocal of 7 is
The modulus notation (| |)
The modulus of a number is the size of that number regardless of its sign. For example |7| is equal to 7, and |− 4| is equal to 4. The modulus of a number is thus never negative.
The factorial symbol (!)
Another commonly used notation is the factorial, denoted by the exclamation mark ‘!’. The number 6!, read ‘six factorial’, or ‘factorial six’, is a shorthand notation for the expression 6×5×4×3×2×1, and the number 8! is shorthand for 8 x 7 × 6 × 5 × 4 × 3 × 2 × 1. Note that 1! equals 1, and by convention 0! is deﬁned as 1 also. Your scientiﬁc calculator is probably able to evaluate factorials of small integers. It is important to note that factorials only apply to positive integers. If n is a positive integer, then n! = n×(n−1)×(n−2)…5×4×3×2×