3. ILLUS. 1. A man sold a horse for $225 and gained $75.
What did the horse cost?
ILLUS. 2. A farmer sold a sheep for m dollars and gained
y dollars. What did the sheep cost? Ans. m – y dollars.
Subtraction is expressed by the sign minus (-).
ILLUS. 3. A man started at a certain point and traveled
north 15 miles, then south 30 miles, then north 20 miles,
then north 5 miles, then south 6 miles. How far is he
from where he started and in which direction?
ILLUS. 4. A man started at a certain point and traveled east x
miles, then west b miles, then east m miles, then east y
miles, then west z miles. How far is he from where he started?
We find a dificulty in solving this last example, because we do not know
just how large x; b; m; y, and z are with reference to each other. This is only one
example of a large class of problems which may arise, in which we find direction
east and west, north and south; space before and behind, to the right and to the
left, above and below; time past and future; money gained and lost; everywhere
these opposite relations. This relation of oppositeness must be expressed in
some way in our representation of numbers.
In algebra, therefore, numbers are considered as increasing from zero in
opposite directions. Those in one direction are called Positive Numbers (or +
numbers); those in the other direction Negative Numbers (or – numbers).
In Illus. 4, if we call direction east positive, then direction west will be nega-
tive, and the respective distances that the man traveled will be +x, -b, +m, +y,
and -z. Combining these, the answer to the problem becomes x-b+m+y-z.
If the same analysis be applied to Illus. 3, we get 15 – 30 + 20 + 5 – 6 = +4, or
4 miles north of starting-point.
The minus sign before a single number makes the number neg-
ative, and shows that the number has a subtractive relation to any
other to which it may be united, and that it will diminish that number
by its value. It shows a relation rather than an operation.
Negative numbers are the second of the two things referred to on page 7, the
introduction of which makes all the difference between arithmetic and algebra.
NOTE.|Negative numbers are usually spoken of as less than zero, because
they are used to represent losses. To illustrate: suppose a man’s money affairs
be such that his debts just equal his assets, we say that he is worth nothing.
Suppose now that the sum of his debts is $1000 greater than his total assets.
He is worse off than by the first supposition, and we say that he is worth less
than nothing. We should represent his property by -1000 (dollars).
1. Express the difference between a and b.