Ch 4: Integers

Question 1

Introducing Integers: Opposites

Answer 1

Every number except zero has an opposite. Opposites are a pair of number that are the same distance from zero on the number line.

All of the following state the same fact:…
4 and -4 are a pair of opposites.
-4 is the opposite of 4.
4 is the opposite of -4.

The set of integers is composed of the counting numbers (1,2,3,4,…) and their opposites, together not negative,

• Negative integers are less than zero
• Zero is an integer that is neither positive nor negative
• Positive integers are greater than zero.

Integers are sometimes called signed numbers, because a “-“ sign is used to indicate a negative number, and a “+” sign is sometimes used to indicate a positive number. In this book, positive numbers are not usually shown with a “+” sign.

Positive and negative integers can be used to show that a measure is greater than zero or less than zero. For example, 5F (Fahrenheit) indicates a temperature 5 degrees greater than zero, and -5F indicates a temperature 5 degrees less than zero.

Another important use for positive and negative integers is to show increases and decreases. For example, a $40 deposit to a bank account could be indicated by +$40, and a $40 withdrawal could use indicated by -40.

You can write a positive number with or without the “+” Positive five can be written.

Question 2

Absolute Values

Answer 2

The absolute value of a number is its distance from zero on the number line. Because absolute value is defined as a distance, it is always non-negative.

Opposite integers have the same absolute value.

The statement |-7| = 7 is read, “the absolute value of negative seven is seven”.

Question 3

Comparing and Ordering

Answer 3

It is possible to compare two or more integers using each integer’s position on the number line as a guide. A lesser number will lie to the left of another number.

Question 4

Adding Integers: Same Signs

Answer 4

To add two or more numbers that have the same sign, add their absolute values. The sum has the same sign as the addends.

It is also helpful to use a number line to add integers. Start at zero. To add a negative number, move left on the number line.

Question 5

Different Signs

Answer 5

To add two numbers that have different signs, subtract the lesser absolute value from the greater absolute value. The sum has the sign of the number with the greater absolute value.

Rules for Addition

• To add numbers that have the same sign, add the absolute values.
• The sum has the same sign as the addends.
• To add two numbers that have different signs, subtract the absolute values. The sum has the sign of the addend with the greater absolute value.

Question 6

Order of Operations

Answer 6

The absolute value symbols act as a set of grouping symbols just as parentheses do.

Question 7

Subtracting Integers

Answer 7

To subtract a number, add its opposite.
Example: Subtract -3 from 5::
5 - (-3) = 5 + 3 = 8

Do not change the sign of the first number in a subtraction problem. Change subtraction to addition, and change the sign of the second number…
- 5 - 2 = - 5 + (-2) = -7
“To subtract a number, add its opposite.”

Question 8

Multiplying Integers

Answer 8

Multiplication of integers can be expressed as repeated addition.
Multiply 3 and -4
You can find the product of 3 and -4 by adding.
(-4) + (-4) + (-4) = -12 = 3 * -4 = -12

Rules for Multiplication

• To multiply two numbers that have the same sign, multiply the absolute values. The product is positive.
• To multiply two numbers that have different signs, multiply the absolute values. The product is negative.
• The product of any number and zero is zero.

Question 9

Dividing Integers

Answer 9

The rules for the sign of a quotient in division are similar to the rules for the sign of a product in multiplication. If two numbers have the same sign, the quotient is positive; if they have different signs, the quotient is negative.

Divide -3,600 by 4
-3,600 / 4 = -900
The numbers have different signs, so the quotient is negative.

Rules for Division

• To divide numbers that have the same sign divide the absolute values.
• The quotient is positive.
• To divide numbers that have different signs, divide the absolute values.
• The quotient is negative.
• Zero divided by any nonzero number is zero; division by zero is undefined.