**Meaning of Negative Numbers**

Let’s assume that you have $3 in your pocket and want to buy a $5 ticket from your friend. You agree to pay $3 now and the rest later, and your friend lets you have the ticket. How much money do you have after the deal? You spent your $3, but you do not have *zero* dollars. You have *less than zero* dollars, because you are $2 in debt, or “in the red.” If you tried to solve this problem by using arithmetic, you would write the problem like this: 3 — 5 = ? But 5 is greater than 3 and so cannot be subtracted from 3. Within arithmetic, there is *no solution* to your problem. However, since this problem is common in practice, there must be a way to solve it mathematically, in a practical way. *Algebra* helps in such a situation by introducing *negative* numbers.

In the problem under discussion, for instance, the result is a value which means *debt*; algebra calls the result “—2,” “negative 2,” or “minus 2.” The number —2 is two less than zero (you owe $2). As you can see, a negative number has a minus sign (—) in front of it. The numbers that you have been using in arithmetic, such as 3, 500, 2/3, 0.057, 2.58, and 1,000,000, are called *positive* numbers. In algebra, positive numbers have a plus sign (+) in front, such as +3, +500, +2/3, +0.057, +2.58, and +1,000,000.

Positive and negative numbers indicate quantities which are opposite or in contrast to each other. Thus, possession, or gain, calls for a positive number; debt, or loss, requires a negative number. There are many other values which are opposites and can be denoted as positive and negative values. A rise in temperature, for example, is indicated by a positive number; a fall in temperature is indicated by a negative number. If a distance traveled east is marked by a positive number, a distance traveled west is marked by a negative number. If, in a game, a player gains 30 points, his score is +30, but if he loses 20 points, his score is —20.

The use of negative and positive numbers greatly increases the possibilities of solving problems that cannot be solved by using positive numbers only, as in arithmetic. The sign + or — indicates the *character* of a number—positive or negative—and the numeral shows its *absolute* value. In the number —4, for example, the character of the number is negative because of the minus sign, and the absolute value is 4.

Positive numbers may be written without the plus sign, as in arithmetic problems. That is, the number +7 may be written as 7, and you should assume that this is a *positive* number with an absolute value 7. Negative numbers, however, *must always* be written with a minus sign in front. The negative number —7, for instance, must have a minus sign; otherwise, it could be taken for a +7. Zero never has any sign, either plus or minus. It is a dividing point between negative and positive numbers.