Page 60 - ArithBook5thEd ~ BCC

Formerly “impossible” subtractions, such as

7 − 12

can now be easily performed.

7 − 12 = 7 + (−12) = −(12 − 7) = −5.

Examples like this show that subtraction is not commutative:

Changing the order of subtraction changes
the result to its opposite.

In symbols, for any two signed numbers A and B,

B − A = −(A − B).

Example 55. 22 − 100 = −78 = −(100 − 22).
We often need to find the difference of two unequal quantities. By convention, difference is given
as a positive quantity. If the two quantities are A and B, their difference is either A − B or B − A
(whichever is positive). Intuitively,

difference = larger − smaller.

Formally, the difference of A and B is defined to be the absolute value of A − B.
Example 56. Find the difference in temperature between −2 Fand 50 F.
◦
◦
Solution. Intuitively,

higher temperature − lower temperature = 50 − (−2) = 52,

◦
so the temperature difference is 52 F. Formally,

| −2 − 50 |=| −52 |=52.

Example 57. The summit of Mount Whitney in California is 14505 feet above sea level. Not far away,
in Death Valley, the lowest point is 282 feet below sea level. What is the altitude difference between
Mount Whitney’s summit and the lowest point in Death Valley?

Solution. Assigning a negative altitude to a point below sea level, the difference is

14505 − (−282) = 14505 + 282 = 14787 feet.

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