6 + 6 + 6 + 6 = 4 · 6A more advanced way to interpret multiplication is as stretching. Thus the number 4 · 6 is 4 times as long as the number 6:
The commutative law of multiplication. This law states that for any numbers a and b, a · b = b · aIn other words the result is the same no matter which number is considered to be multiplied onto which. |
The associative law of multiplication. If we need to multiply three or more numbers together then we do this by multiplying two at a time. The associative law says that the result is the same no matter which two we multiply first. It states that for any numbers a, b and c, (a · b) · c = a · (b · c) |
No matter how complicated the factor a is, 0 · a = 0 |
The number 1 is called the multiplicative identity because multiplying any number a by 1 just gives back a : 1 · a = a |
Moving the − sign around in a product. For any numbers a and b, a · (− b) = (− a) · b = − (a · b) |
The sign of a product having several negative factors. If a product has several negative factors then count how many negative factors there are. If there are an odd number then the product is negative. If there are an even number then the product is positive. |
Definition: Even and odd numbers. The even numbers are the integers …, −6, −4, −2, 0, 2, 4, 6, … and the odd numbers are all the other integers, namely …, −5, −3, −1, 1, 3, 5, …. The even numbers are evenly divisible by 2 and the odd numbers are not. |
In many algebra problems it is important for you to be able to switch the − sign from one role to another.
The expression What the − sign means in this expression − 5 This is just a number to the left of 0 on the number line. 0 + (− 5) This is an addition. We are adding −5 onto zero. 0 − 5 This is a subtraction. We are subtracting 5 from zero. −1 · 5 This is a multiplication. We are multiplying 5 by −1. − ( 5 ) This means “the opposite of 5”.
Algebra Coach Exercises |
Multiplication of expressions,
Multiplication of algebraic fractions.