Chapter Notes: Integers: Addition And Subtraction
Imagine standing on a number line. You can walk forward, or you can walk backward. You can climb up a hill, or you can go down into a valley. Positive and negative numbers behave in similar ways. When we add and subtract integers-whole numbers that can be positive, negative, or zero-we are exploring how quantities combine and compare. Understanding how to add and subtract integers helps us solve real-world problems like temperature changes, bank account balances, elevation differences, and game scores. In this chapter, we will learn the rules for combining integers and see how the number line helps us visualize these operations.
What Are Integers?
Integers are the set of whole numbers and their opposites. This set includes all positive whole numbers (1, 2, 3, ...), all negative whole numbers (-1, -2, -3, ...), and zero. Integers do not include fractions or decimals. We use integers to represent quantities that can increase or decrease from a starting point, like floors in a building where the ground floor is zero, floors above ground are positive, and basement levels are negative.
On a number line, integers are evenly spaced. Zero sits in the middle. Positive integers extend to the right, and negative integers extend to the left. The farther right you go, the greater the number. The farther left you go, the smaller the number. For example, 5 is greater than 2, and -2 is greater than -5.
Understanding Opposites and Absolute Value
Every integer has an opposite. The opposite of a number is the same distance from zero but on the other side of the number line. The opposite of 7 is -7. The opposite of -3 is 3. The opposite of 0 is 0 itself. When we add a number and its opposite, the result is always zero.
The absolute value of an integer is its distance from zero on the number line, without considering direction. Distance is always positive or zero. We write absolute value with two vertical bars around the number. For example:
- \( |5| = 5 \) because 5 is five units away from zero.
- \( |-5| = 5 \) because -5 is also five units away from zero.
- \( |0| = 0 \) because zero is zero units away from itself.
Absolute value helps us understand the size of a number without worrying about its sign. When we add or subtract integers, absolute value plays an important role in determining the result.
Adding Integers with the Same Sign
When we add two integers that have the same sign-both positive or both negative-we follow a simple rule: add their absolute values and keep the common sign.
Adding Two Positive Integers
Adding two positive integers is just like adding whole numbers. You combine the quantities and the result is positive.
Example: Find the sum of 8 and 5.
What is 8 + 5?
Solution:
Both numbers are positive, so we add their values directly.
8 + 5 = 13
The sum is 13.
Adding Two Negative Integers
When adding two negative integers, think of combining debts or going farther down from zero. Add their absolute values, then place a negative sign in front of the result.
Example: Find the sum of -6 and -4.
What is -6 + (-4)?
Solution:
Both integers are negative. First, find the absolute values: \( |-6| = 6 \) and \( |-4| = 4 \).
Add the absolute values: 6 + 4 = 10.
Since both numbers are negative, the sum is negative: -10.
The sum of -6 and -4 is -10.
Think of it this way: if you owe $6 and then you owe another $4, you owe a total of $10, which we represent as -10.
Adding Integers with Different Signs
When we add two integers with different signs-one positive and one negative-we need to find the difference between their absolute values. The sign of the result matches the sign of the integer with the larger absolute value.
Step-by-Step Process
- Find the absolute value of each integer.
- Subtract the smaller absolute value from the larger absolute value.
- The result takes the sign of the integer with the larger absolute value.
Example: Add 9 and -5.
What is 9 + (-5)?
Solution:
Find the absolute values: \( |9| = 9 \) and \( |-5| = 5 \).
Subtract the smaller from the larger: 9 - 5 = 4.
Since 9 has the larger absolute value and is positive, the result is positive.
The sum is 4.
Example: Add -12 and 7.
What is -12 + 7?
Solution:
Find the absolute values: \( |-12| = 12 \) and \( |7| = 7 \).
Subtract the smaller from the larger: 12 - 7 = 5.
Since -12 has the larger absolute value and is negative, the result is negative.
The sum is -5.
Imagine the temperature is 7 degrees above zero, then it drops by 12 degrees. You end up 5 degrees below zero, or -5°F.
Subtracting Integers Using Addition
Subtraction of integers can sometimes feel tricky, but there is a powerful rule that makes it simple: subtracting an integer is the same as adding its opposite. This means every subtraction problem can be rewritten as an addition problem.
The rule is:
\[ a - b = a + (-b) \]Here, \( a \) is the first integer, and \( b \) is the integer being subtracted. To subtract \( b \), we add the opposite of \( b \), which is \( -b \).
Rewriting Subtraction as Addition
Follow these steps:
- Keep the first number exactly as it is.
- Change the subtraction sign to an addition sign.
- Replace the second number with its opposite.
- Use the rules for adding integers.
Example: Subtract 6 from 10.
What is 10 - 6?
Solution:
Rewrite as addition: 10 - 6 = 10 + (-6).
Now add: \( |10| = 10 \) and \( |-6| = 6 \).
Subtract the smaller absolute value from the larger: 10 - 6 = 4.
Since 10 is larger and positive, the result is positive: 4.
The result is 4.
Example: Subtract 8 from 3.
What is 3 - 8?
Solution:
Rewrite as addition: 3 - 8 = 3 + (-8).
Find the absolute values: \( |3| = 3 \) and \( |-8| = 8 \).
Subtract: 8 - 3 = 5.
Since -8 has the larger absolute value and is negative, the result is negative: -5.
The result is -5.
Subtracting a Negative Integer
Subtracting a negative integer is the same as adding a positive integer. This follows directly from the rule that subtracting is the same as adding the opposite. The opposite of a negative number is a positive number.
For example:
\[ 5 - (-3) = 5 + 3 = 8 \]When you subtract a negative, you move to the right on the number line, just like when you add a positive number.
Example: Subtract -7 from 2.
What is 2 - (-7)?
Solution:
Rewrite as addition: 2 - (-7) = 2 + 7.
Add the two positive integers: 2 + 7 = 9.
The result is 9.
Example: Subtract -10 from -4.
What is -4 - (-10)?
Solution:
Rewrite as addition: -4 - (-10) = -4 + 10.
Find absolute values: \( |-4| = 4 \) and \( |10| = 10 \).
Subtract: 10 - 4 = 6.
Since 10 has the larger absolute value and is positive, the result is positive: 6.
The result is 6.
Think of it this way: if you have a debt of $4 and someone cancels a debt of $10, you actually gain $6.
Using the Number Line for Addition and Subtraction
The number line is a powerful visual tool for understanding integer operations. When adding a positive integer, move to the right. When adding a negative integer, move to the left. Subtraction can be thought of as adding the opposite, so subtracting a positive means moving left, and subtracting a negative means moving right.
Visualizing Addition
To compute \( -3 + 5 \) on a number line:
- Start at -3.
- Since you are adding 5 (a positive number), move 5 units to the right.
- You land on 2.
So, \( -3 + 5 = 2 \).
Visualizing Subtraction
To compute \( 4 - 7 \) on a number line:
- Start at 4.
- Rewrite as \( 4 + (-7) \), which means move 7 units to the left.
- You land on -3.
So, \( 4 - 7 = -3 \).
Combining Multiple Operations
Real problems often involve more than one addition or subtraction. We solve these step by step, working from left to right, or by first converting all subtractions to additions of opposites and then combining like signs.
Example: Simplify the expression: 8 - 12 + 5 - (-3).
What is the value of the expression?
Solution:
Rewrite all subtractions as additions:
8 - 12 + 5 - (-3) = 8 + (-12) + 5 + 3Now combine from left to right:
First: 8 + (-12) = -4
Next: -4 + 5 = 1
Finally: 1 + 3 = 4
The value of the expression is 4.
Example: A submarine is at a depth of 150 meters below sea level.
It descends another 75 meters, then rises 200 meters.What is the submarine's final depth?
Solution:
Represent depth below sea level as negative. Start at -150 meters.
Descending 75 meters means subtracting 75: -150 - 75 = -150 + (-75) = -225.
Rising 200 meters means adding 200: -225 + 200.
Find absolute values: \( |-225| = 225 \) and \( |200| = 200 \).
Subtract: 225 - 200 = 25.
Since -225 has the larger absolute value, the result is negative: -25.
The submarine's final depth is 25 meters below sea level, or -25 meters.
Properties of Integer Addition
Integer addition follows several important properties that help simplify calculations and solve problems more flexibly.
Commutative Property
The commutative property states that changing the order of the integers does not change the sum:
\[ a + b = b + a \]For example: \( 7 + (-3) = (-3) + 7 = 4 \).
Associative Property
The associative property states that when adding three or more integers, the way we group them does not change the sum:
\[ (a + b) + c = a + (b + c) \]For example: \( (5 + (-2)) + 8 = 5 + ((-2) + 8) \).
Both sides simplify to 11.
Identity Property
The identity property states that adding zero to any integer gives the same integer:
\[ a + 0 = a \]Zero is called the additive identity.
Inverse Property
The inverse property states that every integer has an opposite (called its additive inverse) such that their sum is zero:
\[ a + (-a) = 0 \]For example: \( 9 + (-9) = 0 \) and \( -15 + 15 = 0 \).
Real-World Applications
Integer addition and subtraction appear in many everyday contexts. Understanding these operations allows us to solve practical problems involving money, temperature, elevation, time zones, and more.
Temperature Changes
Temperature can rise or fall, and we use integers to represent these changes. If the temperature is -8°F in the morning and rises by 15°F during the day, the new temperature is \( -8 + 15 = 7 \)°F.
Bank Account Balances
Money deposited into an account is positive, and money withdrawn or spent is negative. If your account has $45 and you spend $60, your balance becomes \( 45 - 60 = 45 + (-60) = -15 \), meaning you are overdrawn by $15.
Elevation and Depth
Elevation above sea level is positive, and depth below sea level is negative. A diver at 30 meters below sea level who swims up 12 meters is at \( -30 + 12 = -18 \) meters, or 18 meters below sea level.
Football Yardage
In football, gaining yards is positive and losing yards is negative. A team gains 8 yards, then loses 5 yards, then gains 12 yards. The net yardage is \( 8 + (-5) + 12 = 8 - 5 + 12 = 15 \) yards.
Example: The temperature at sunrise was -6°C.
By noon, it increased by 9°C.
By evening, it dropped by 4°C.What was the temperature in the evening?
Solution:
Start with -6°C.
Add the increase: -6 + 9 = 3°C at noon.
Subtract the drop: 3 - 4 = 3 + (-4) = -1°C in the evening.
The evening temperature was -1°C.
Common Mistakes and How to Avoid Them
Students sometimes make errors when adding and subtracting integers. Being aware of these common mistakes can help you avoid them.
Mistake 1: Forgetting to Change Signs When Subtracting
When subtracting a negative number, remember to change the operation to addition and change the sign of the number being subtracted. For example, \( 5 - (-2) \) is not \( 5 - 2 \); it is \( 5 + 2 = 7 \).
Mistake 2: Confusing Absolute Value with the Number Itself
Absolute value strips away the sign. Don't forget to reapply the correct sign after calculating. For instance, when adding -8 and -3, the absolute values are 8 and 3, which sum to 11, but since both are negative, the result is -11, not 11.
Mistake 3: Incorrect Order of Operations
When combining multiple operations, work from left to right unless grouping symbols tell you otherwise. Don't skip steps or try to do too much at once in your head.
Mistake 4: Misreading the Number Line
Always double-check which direction you are moving. Adding a negative means moving left; subtracting a negative means moving right.
Summary of Rules
With practice, adding and subtracting integers becomes second nature. The key is understanding the rules, visualizing with the number line, and carefully keeping track of signs. Integer operations are foundational skills that you will use throughout algebra, geometry, science, and everyday problem-solving.
