Some important properties of real numbers are listed below.
Let x, y and z be any three real numbers. Then,
If two numbers x and y are both positive, then (x + y) and xy are positive.
For example, 3 + 2.1 = 5.1 and (3) (2.1) = 6.3
If two numbers x and y are both negative, then (x + y) is negative and xy is positive.
For example, (−3) + (−2.1) = −5.1 and (−3) (−2.1) = 6.3
If one of two numbers x and y is positive and the other is negative, then (x + y) has the sign of the number larger in absolute value and xy is negative.
For example, (−3) + 2.1 = −0.9 and (−3) (2.1) = −6.3
If xy = 0, then x = 0 and/or y = 0.
For example, (a + 3) (a − 4) = 0 ⇒ a + 3 = 0 or a − 4 = 0 ⇒ a = −3 or a = 4
|x + y| ≤ |x| + |y|
For example, the equality holds when x = 9 and y = 4 because |x + y| = |9 + 4| = |13| = 13 and |x| + |y| = |9| + |4| = 9 + 4 = 13. On the other hand, the inequality holds when x = 9 and y = −4 because |x + y| = |9 + (−4)| = |5| = 5 and |x| + |y| = |9| + |−4| = 9 + 4 = 13.
x + y = y + x [Commutative property of addition]
For example, 3.5 + 5.1 = 5.1 + 3.5 = 8.6
Since the order in which numbers are added is immaterial, addition is commutative.
Note that subtraction is not commutative because x − y is not necessarily equal to y − x.
xy = yx [Commutative property of multiplication]
For example, (3.5) (5) = (5) (3.5) = 16.5
Since the order in which numbers are multiplied is immaterial, multiplication is commutative.
Note that division is not commutative because x / y is not necessarily equal to y / x.
(x + y) + z = x + (y + z) [Associative property of addition]
For example, (3.5 + 5.1) + 1.2 = 8.6 + 1.2 = 9.8 and 3.5 + (5.1 + 1.2) = 3.5 + 6.3 = 9.8
The parenthesis simply group the numbers to show which two numbers are added first. Since the same answer is obtained irrespective of the grouping, addition is associative.
(xy) z = x (yz) [Associative property of multiplication]
For example, (3 × 5) × 1.1 = (15) 1.1 = 16.5 and 3 × (5 × 1.1) = 3 (5.5) = 16.5
The parenthesis simply group the numbers to show which two numbers are multiplied first. Since the same answer is obtained irrespective of the grouping, multiplication is associative.
The associative property of multiplication may be sometimes useful for quick calculations.
For example, (7 √2) √18 = 7 (√2 √18) = 7 (√36) = 7 (6) = 42
x (y + z) = xy + xz [Distributive property of multiplication over addition]
For example, 3 (5 + 1.1) = 3 (6.1) = 18.3 and (3 × 5) + (3 × 1.1) = 15 + 3.3 = 18.3
Addition can be performed first followed by multiplication, or multiplication can be performed first followed by addition. Since the same answer is obtained either way, multiplication is distributive over addition.
The distributive property of multiplication over addition may be sometimes useful for quick calculations.
For example, 6.93(2.7) + 6.93(7.3) = 6.93 (2.7 + 7.3) = 6.93 (10) = 69.3
Example
A store made a profit of $220.50 on Monday, a loss of $25.75 on Tuesday, a loss of $36.50 on Wednesday, a profit of 97.25 on Thursday, a profit of $395 on Friday, and a profit of $623.75 on Saturday. The store was closed on Sunday. What was the total profit (or loss) the store made during the week? Solution.
Total = 220.50 + (−25.75) + (−36.50) + 197.25 + 395 + 623.75
Adding the profits, 220.50 + 197.25 + 395 + 623.75 = 1436.50
Adding the losses, 25.75 + 36.50 = 62.25
Total = Profits − Losses = 1436.50 − 62.25 = 1374.25
Thus, the total profit the store made during the week is 374.25
Example
(−8) (¾) (−7) (−¼) (−1.1) Solution.
Multiply the first two factors, then multiply the result by the third factor, and so on.
(−8) (¾) = −6
(−6) (−7) = 42
(42) (−¼) = −10.5
(−10.5) (−1.1) = 11.55
So, (−8) (¾) (−7) (−¼) (−1.1) = 11.55
Note that the sign of the product (or quotient) is positive when there are no negative factors or an even number of negative factors.
In this example, there are 4 negative factors and so the product is positive.
The sign of the product (or quotient) is negative when there are an odd number of negative factors.
