Lesson 2 · from §12 of the 1910 text
Signed Numbers in Real Life
The number line can measure many everyday things. In fact, people already use it without calling it that.
A thermometer picks one position of the mercury as zero and marks degrees both above and below it. A reading of 18° means nothing until you say above zero or below zero. Usually +18° means above and -18° means below.
Latitude works the same way. Every point on the equator has latitude zero. Latitude 40° N means 40° north of the equator and 30° S means 30° south of it, so +40° and -30° say the same thing.
Money too: property you own can be positive and debts negative, and adding them up tells you where you really stand.
Quick check
The temperature is -4° and it rises 9°. What is the new temperature?
From the history of algebra
Assets and debts. The earliest known explanation of positive and negative numbers used exactly this illustration of property and debts. It appears in the writings of the Hindus before 700 A.D., long before negative numbers were accepted as meaning anything at all. With this illustration the Hindus were nearly a thousand years ahead of their time.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
If the temperature is now +10°, what will it be after a fall (a) of 5°? (b) of 10°? (c) of 18°?
If the temperature is now -12°, what will it be after a rise (a) of 7°? (b) of 12°? (c) of 25°?
In the preceding exercise, change rise to fall. What is the temperature after a fall (a) of 7°? (b) of 12°? (c) of 25°?
A ship sails south from latitude +13° to latitude -7°. If one degree is 69 miles, how far did it sail?
A ship sails south from latitude +20° at the rate of 4° daily. In what latitude is it at the end of each of the next 6 days? In how many days will it reach latitude -16°?
A man's property is worth $5200 and his debts amount to $2300. Using positive and negative numbers, represent (a) his property, (b) his debts, and (c) his financial standing.
Reverse the debts and property in Exercise 6: property $2300 and debts $5200. Represent (a) his property, (b) his debts, and (c) his financial standing.
The temperature at 6:00 A.M. was -12°. During the morning it rose 3° an hour. What was the temperature at 9:00 A.M.? at 10:00 A.M.? at 1:00 P.M.?