Topic 6: INTEGERS (Form 1)

INTEGERS Integers are whole numbers that have postive sign (+)  or negative sign (-), including zero. Addition of Integer

Problem:	Kamal owes his friend Zainab RM3. If he borrows another RM6, how much will he owe her altogether?
Solution:	This problem is quite simple: just add RM3 and RM6 and the result is RM9.

The problem above can be solved using addition of integers. Owing RM3 can be represented by -3 and owing RM6 can be represented by -6. The problem becomes: -3 + -6 = -9

Look at the number line below. If we start at 0, and move 3 to the left, we land on -3. If we then move another 6 to the left, we end up at -9.

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Rule:   The sum of two negative integers is a negative integer.

Example 1:	Find the sum of each pair of integers. You may draw a number line to help you solve this problem.
	Adding Negative Integers Integers Sum (-2) + (-9) = -11 (-5) + (-8) = -13 (-13) + (-17) = -20

Do not confuse the sign of the integer with the operation being performed. Remember that:

(-2) + (-9) = -11 is read as Negative 2 plus negative 9  equals negative 11.

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Rule:   The sum of two positive integers is a positive integer.

Example 2:	Find the sum of each pair of integers. You may draw a number line to help you solve this problem.
	Adding Positive Integers Integers Sum (+2) + (+9)= +11 (+17) + (+5) +22 (+29) + (+16) +45

Do not confuse the sign of the integer with the operation being performed. Remember that:

(+29) + (+16) = +45 is read as Positive 29 plus positive 16  equals positive 45.

So far we have added integers with like signs (either both negative or both positive). What happens when we add integers with unlike signs? How do we add a positive and a negative integer, or a negative and a positive integer?

Procedure:	To add a positive and a negative integer (or a negative and a positive integer), follow these steps:
	1.  Find the absolute value of each integer. 2. Subtract the smaller number from the larger number you get in Step 1. 3. The result from Step 2 takes the sign of the integer with the greater absolute value.

The absolute value of an integer is its distance from zero on the number line.
For example, |^-3| = 3 and |^-5| = 5.

We will use the above procedure to add integers with unlike signs in Examples 3 through 7. Refer to the number line to help you visualize the process in each example. We will use money as an alternative method for adding integers.

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Example 3:	Find the sum of (+7) + (-4)
	Step 1:   |+7| = 7 and |-4| = 4
	Step 2:   7 - 4 = 3
	Step 3:   The number 3 will take a positive sign since +7 is farther from zero than -4.
Solution 1:	(+7) + (-4) = +3
Solution 2:	If you start with RM7 and you owe RM4, then you end up with RM3.

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Example 4:	Find the sum of -9 and +5.
	Step 1:   |-9| = 9 and |+5| = 5
	Step 2:   9 - 5 = 4
	Step 3:   The number 4 will take a negative sign since -9 is farther from 0 than +5.
Solution 1:	-9 + +5 = -4
Solution 2:	If you owe RM9 and you are paid RM5, then you are still short RM4.

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Example 5:	Find the sum of +6 and -7.
	Step 1:   |+6| = 6 and |-7| = 7
	Step 2:   7 - 6 = 1
	Step 3:   The number 1 will take a negative sign since -7 is farther from 0 than +6.
Solution 1:	+6 + -7 = -1
Solution 2:	If you start with RM6 and you owe RM7, then you are still short RM1.

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Example 6:	Find the sum of -6 and +7.
	Step 1:   |-6| = 6 and |+7| = 7
	Step 2:   7 - 6 = 1
	Step 3:   The number 1 will take a positive sign since +7 is farther from 0 than -6.
Solution 1:	-6 + +7 = +1
Solution 2:	If you owe RM6 and you are paid RM7, then you end up with RM1.

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Example 7:	Find the sum of +9 and -9.
	Step 1:   |+9| = 9 and |-9| = 9
	Step 2:   9 - 9 = 0
	Step 3:   The integer 0 has no sign.
Solution 1:	+9 + -9 = 0
Solution 2:	If you start with RM9 and you owe RM9, then you end up with RM0.

In Example 7 you will notice that the integers ⁺9 and ^-9 are opposites. Look at the problems below. Do you see a pattern?

-100 + +100 = 0

+349 + -349 = 0

-798 + +798 = 0

Rule:   The sum of any integer and its opposite is equal to zero.

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Summary:	Adding two positive integers always yields a positive sum; adding two negative integers always yields a negative sum. To find the sum of a positive and a negative integer, take the absolute value of each integer and then subtract these values. The result takes the sign of the integer with the larger absolute value. The sum of any integer and its opposite is equal to zero.
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Subtraction of Integer


Problem: The temperature in Anchorage, Alaska was 8°F in the morning and dropped to^-5°F in the evening. What is the difference between these temperatures?

Solution: We can solve this problem using integers. Using the number line below, the distance from ⁺8 to 0 is 8, and the distance from 0 to ^-5 is 5, for a total of 13.

(+8) - (-5) =  +13. The difference is 13 degrees.

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Problem: The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the difference between these two elevations?


Solution: When solving problems with large integers, it is not always practical to rely on the number line. Using integer arithmetic this problem becomes:
(+20,320) - (-282) = ?

We need a rule for subtracting integers in order to solve this problem.

Rule:   To subtract an integer, add its opposite.

The opposite of -282 is +282, so we get: (+20,320) - (-282) = (+20,320) + (+282) = +20,602

In the above problem, we added the opposite of the second integer and subtraction was transformed into addition. Let's look at some simpler examples of subtracting integers.

Example 1: (+5) - (+2)

  Step 1: The opposite of +2 is -2

  Step 2: Subtraction becomes addition

  Step 3: (+5) - (+2) becomes (+5) + (-2) = +3

Example 2:  Find the difference between each pairs of integers.

Subtracting Integers
Subtract	Add The Opposite	Result
+9 - +4 =	+9 + -4 =	+5
+9 - -4 =	+9 + +4 =	+13
-9 - +4 =	-9 + -4 =	-13
-9 - -4 =	-9 + +4 =	-5

Notice that in each problem above, the first integer remained unchanged. Also, do not confuse the sign of the integer with the operation being performed. Remember that:
^-9 + ⁺4 = ^-5   is read as Negative 9 plus positive 4 equals negative 5.

SUMMARY:

To subtract an integer, add its opposite. When subtracting integers, it is important to show all work, as we did in this lesson. If you skip steps, or do the work in your head, you are very likely to make a mistake--even if you are a top math student!