2.4 Properties of integers
The associative properties of operations with integers
Multiplication of whole numbers is associative. The associative property of multiplication means that in a product with several factors, the factors can be placed in any sequence, and the calculations can be performed in any sequence. For example, the following sequences of calculations will all produce the same answer:
- \(2 \times 3\), the answer of \(2 \times 3\) multiplied by \(5\), the new answer multiplied by \(10\)
- \(2 \times 5\), the answer of \(2 \times 5\) multiplied by \(10\), the new answer multiplied by \(3\)
- \(10 \times 5\), the answer of \(10 \times 5\) multiplied by \(3\), the new answer multiplied by \(2\)
- \(3 \times 5\), the answer of \(3 \times 5\) multiplied by \(2\), the new answer multiplied by \(10\)
The answer in each case is \(300\). Check this for yourself!
- associative property of multiplication
- in a product with several factors, the factors can be placed in any sequence, and the calculations can be performed in any sequence
Multiplication with integers is also associative.
- \(- 2 \times ( - 3) \times 5 \times 10 = (6) \times 5 \times 10 = 30 \times 10 = 300\)
- \(5 \times ( - 2) \times ( - 3) \times 10 = ( - 10) \times ( - 30) = 300\)
Addition with integers is associative. Calculating the following illustrates this property:
- \(80 - 30 + 40 - 20 = 50 + 40 - 20 = 90 - 20 = 70\)
- \(80 + ( - 30) + 40 + ( - 20) = (80 - 30) + (40 - 20) = 50 + 20 = 70\)
- \(- 30 + 80 - 20 + 40 = ( - 30 + 80 - 20) + 40 = (80 - 50) + 40 = 30 + 40 = 70\)
- \(( - 30) + 80 + ( - 20) + 40 = ( - 30 - 20) + (80 + 40) = ( - 50) + (120) = ( - 50) + (50 + 70) = 70\)
- \(- 20 - 30 + 40 + 80 = - 50 + 40 + 80 = - 10 + 80 = 70\)
When rearranging integers in addition, you need to be careful not to leave the minus sign behind. For example, \(20 - 30 = 20 + ( - 30) = - 30 + 20 = - 10\).
Additive and multiplicative inverses for integers
You probably agree with the following statements:
\[5 + ( - 5)\ = \ 0\] \[10 + ( - 10)\ = \ 0\] \[20 + ( - 20) = 0\]We may say that for each “positive” number, there is a corresponding or opposite negative number. Two positive and negative numbers that correspond, for example 3 and (−3), are called additive inverses. They cancel each other out when you add them.
When you add any number to its additive inverse, the answer is \(0\) (the additive property of \(0\)). For example, \(120 + ( - 120) = 0\).
The multiplicative inverse of a number is a number that when multiplied by the original number, gives the product of \(1\). For example, the multiplicative inverse of \(5\) is \(\frac{1}{5}\), because \(5 \times \frac{1}{5} = 1\).
When the product of two numbers is \(1\), the numbers are multiplicative inverses of each other. These numbers can also be called the “reciprocals” of one another. For example, \(- \frac{1}{2}\) is the reciprocal of \(- 2\).