Arithmetic

A

Adding Negative Integers

By referring once again to the number line, which includes both positive and negative numbers, we can see that the sum of any two negative numbers is negative and equals the total of the absolute values of the two numbers, but negative:

If, however, the signs of the two numbers are opposite, then we must proceed more cautiously.

To add -4 and 2, for example, begin at -4 and add 2 by moving 2 units forward, in the positive direction to the right. The result is -2. The answer is the difference between the absolute values of the two numbers, with the result taking the sign of the larger number. The answer to this problem must be negative because in terms of absolute value, |-4| > |2|. (The symbol > means “greater than.”) We obtain the answer, however, by calculating the difference between the absolute values 4 and 2. Similarly, to add -3 and 7, begin at -3 and move 7 units forward, reaching 4. Two rules simplify this process:

Rule 1: To add two numbers with the same sign, add the absolute values of the two numbers, then give the answer the sign of the numbers in question:

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Rule 2: To add two numbers with different signs, determine their absolute values and subtract the smaller number from the larger. Give the result the sign of the number with the larger absolute value:

B

Subtracting Negative Integers

The subtraction of negative numbers is easy to visualize on the number line. Subtracting one number from another involves moving along the number line in the opposite direction from addition. To subtract 5 from 7, for example, begin at 7 and move in the negative direction 5 units, reaching 2 as the answer. Similarly, to subtract -2 from 5, start at 5 but reverse direction and move 2 units in the positive direction to 7. Subtracting -2 basically means adding 2, or more generally, subtracting a negative number is the same thing as adding the absolute value of the number. We can therefore change two negative (or minus) signs in a row into a single positive (or plus) sign. For example:

We can change a negative and a positive sign together into a negative sign:

C

Multiplying Negative Integers

Determining the sign of a product is straightforward. To multiply two numbers with the same sign, multiply their absolute values and give the resulting product a positive sign:

To multiply two numbers with different signs, multiply their absolute values and give the resulting product a negative sign:

D

Dividing Negative Integers

Division is the inverse of multiplication; therefore, dividing positive or negative numbers involves rules similar to those for multiplication. To divide two numbers with the same sign, divide their absolute values and give the resulting quotient a positive sign:

To divide two numbers with different signs, divide their absolute values and give the resulting quotient a negative sign:

IV

Operations With Fractions

Numbers that represent parts of a whole are called fractions or rational numbers. Simple fractions are familiar: a dime is s of a dollar; ‚ of a pie plus y of a pie is • of a pie; and so on. In general, we can express fractions as the quotient of two integers a and b:

The top number in a fraction is called the numerator and the bottom number is called the denominator. Two types of fractions exist: proper and improper. A proper fraction is one in which the numerator is smaller than the denominator; ’, -, and  are all proper fractions. An improper fraction is one in which the numerator is larger than the denominator; ”, -ž, and œ are improper fractions. We can convert improper fractions to mixed fractions or whole numbers (for example, ” = 1y, -ž = -2, and œ = 2€) by dividing the numerator by the denominator and expressing any remainder as a fraction of the denominator.

A fraction is said to be reduced to lowest terms if neither the numerator nor the denominator has a factor in common. A factor is a number by which another number can be divided evenly. For example, › is not reduced to lowest terms because both 6 and 8 have 2 as a factor:

Since a number divided by itself is always equal to 1, = 1. Multiplying any number by 1 does not change the number, so × • = •. Reduced to lowest terms, then, › is •.

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