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1-5 Addition of Real Numbers
The result of adding two or more numbers is the sum of the numbers.
Gains and losses of money can help in finding sums of positive and negative numbers.
Example: You find $17 and later you make a purchase for $13. How much
money did you gain/lose?
Finding money increases the amount you have (+17 or 17). A purchase
decreases the amount you have (–13). You still have $4 of the $17 you
found so you have a gain of $4.
17 + (–13) = 4
Example: You lost your wallet with $30 in it. Later, you borrowed $5 for
lunch. How much money did you gain/lose?
Both events decrease the amount of money you have. After losing the
wallet (–30) and borrowing money (–5), you have a loss of $35 (–35).
–30 + (–5) = –35
----------------------------------------------------------------------------------------------------------Adding with a Number Line (p. 42)
You can use the number line to help visualize the addition of real numbers.
To find a + b using a number line,
1. start at a
2a. if b is positive, move b units to the right
b. if b is negative, move b units to the left
c. if b is zero, stay at a
3. the number where you end up is the sum of a and b
Example 2 (p. 42): Add 3 + (–5).
1. start at 3 on the number line
2. the second number is negative; move 5 units to the left
<-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
–4 –3 –2 –1 0 1 2 3 4
|<--------------------------|
3.
you end up at –2; 3 + (–5) = –2
Example: Add –3 + (–4).
1. start at –3 on the number line
2. the second number is negative; move 4 units to the left
<-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
–8 –7 –6 –5 –4 –3 –2 –1 0
|<---------------------|
3.
you end up at –7; –3 + (–4) = –7
Example: Add –2 + 6.
1. start at –2 on the number line
2. the second number is positive; move 6 units to the right
<-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
–3 –2 –1 0 1 2 3 4 5
|-------------------------------->|
3. you end up at 4; (–2) + 6 = 4
----------------------------------------------------------------------------------------------------------The Number Line and Properties of Addition
The number line can be used to picture some useful properties of addition.
Law of Opposites for Addition (1.6 on p. 49)
Numbers that differ in sign but have the same absolute value are called opposites or
additive inverses. Examples are 3 and –3, –5 and 5.
The sum of any real number and its opposite is zero.
𝑎 + (– 𝑎) = 0
Example: Add –3 + 3.
1. start at –3 on the number line
2. the second number is positive; move 3 units to the right
<-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
–4 –3 –2 –1 0 1 2 3 4
|--------------->|
3.
you end up at 0; –3 + 3 = 0
Identity Property of Zero for Addition (p. 43)
When you add zero to a number or a number to zero the sum is the same as the number.
Zero is called the identity element for addition.
a+0=a
0+a=a
----------------------------------------------------------------------------------------------------------Adding without a Number Line (p. 43)
If the numbers have the same sign
 add the absolute values
 use the common sign as the sign of the sum
The sum of two positive numbers is always positive.
The sum of two negative numbers is always negative.
Example: Find the sum of each pair of numbers.
8+9
The sum is 17
–11 + (–15)
The sum is –26
If the numbers have different signs
 subtract the smaller absolute value from the larger absolute value
 use the sign of the number with the larger absolute value as the sign of the sum
Examples: Find the sum of each pair of numbers.
–4 + 9 = 5
Difference is 5. Number with larger abs is positive.
–15 + 8 = –7
Difference is 7. Number with larger abs is negative.
6 + (–10) = –4
Difference is 4. Number with larger abs is negative.
12 + (–3) = 9
Difference is 9. Number with larger abs is positive.
----------------------------------------------------------------------------------------------------------Applications
Positive and negative numbers are used in everyday life to represent such things as gains
and losses in the stock market, rising and falling temperatures, deposits and withdrawals
on bank statements and ascending and descending motion.
Some applications involve the sum of several real numbers. If you have to add several
positive and negative numbers, you have a choice of methods.
Method 1: Apply the Order of Operations and add the numbers from left to right. This
may result in several additions involving numbers with different signs.
Method 2: Apply the Commutative and Associative Properties to add all the positive
numbers, then add all the negative numbers. Finally, add the positive sum and the
negative sum. This method results in only one addition of numbers with different signs.
Example: A glider is released at an altitude of 1,000 feet. It immediately drops 70 feet
but finds an updraft and climbs 2,100 feet. It looses the updraft and descends 230 feet but
then finds another updraft and climbs 1,200 feet. The following sum expresses the
altitude of the glider following these rises and falls in altitude.
1,000 + (–70) + 2,100 + (–230) + 1,200
Method 1
1,000 + (–70) + 2,100 + (–230) + 1,200
930 + 2,100 + (–230) + 1,200
3,030 + (–230) + 1,200
2,800 + 1,200
4,000
Method 2
1,000 + (–70) + 2,100 + (–230) + 1,200
(1,000 + 2,100 + 1,200) + (–70 + (–230))
4,300 + (–300)
4,000
The altitude of the glider is 4,000 meters.
Also see Example 6 on page 44 and Example 7 on page 45.
----------------------------------------------------------------------------------------------------------Combining Like Terms (p. 45)
Use the rules for adding real numbers when combining like terms.
Examples: Simplify each expression.
−11𝑥 + 7𝑥 = (−11 + 7)𝑥 = −4𝑥
7𝑦 + (−12𝑧) + (−9𝑦) + 15𝑧
= −2𝑦 + 3𝑧
[7 + (−9)]𝑦 = −2𝑦
3(7𝑥 + 8) + (−25𝑥) = 21𝑥 + 24 + (−25𝑥)
= −4𝑥 + 24
[−12 + 15]𝑧 = 3𝑧
[21 + (−25)]𝑥 = −4𝑥