Chapter Notes: Rational Numbers: Addition And Subtraction
Rational numbers are numbers that can be expressed as fractions, including whole numbers, fractions, and their opposites. When you work with money, measure ingredients for recipes, or track temperature changes, you're often adding and subtracting rational numbers. Understanding how to add and subtract these numbers-whether they're positive or negative, whole numbers or fractions-is an essential skill that connects arithmetic to real-world problem solving. In this chapter, you'll learn strategies for combining rational numbers accurately and efficiently.
Understanding Rational Numbers
A rational number is any number that can be written as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \) is not zero. This definition includes many types of numbers you already know.
- Whole numbers like 5 can be written as \( \frac{5}{1} \)
- Fractions like \( \frac{3}{4} \) are already in rational form
- Negative numbers like -7 can be written as \( \frac{-7}{1} \)
- Decimals that terminate like 0.25 equal \( \frac{1}{4} \)
- Decimals that repeat like 0.333... equal \( \frac{1}{3} \)
Rational numbers can be positive, negative, or zero. They appear everywhere: bank account balances can be positive (deposits) or negative (overdrafts), elevations can be above or below sea level, and temperatures can be above or below freezing.
Adding Rational Numbers with the Same Sign
When you add two rational numbers that have the same sign-both positive or both negative-the process is straightforward.
Adding Two Positive Rational Numbers
When both numbers are positive, simply add their absolute values. The result is positive.
Example: A bakery sells 2.5 pounds of cookies in the morning.
In the afternoon, they sell 3.75 pounds more.How many pounds did they sell in total?
Solution:
Both quantities are positive, so we add their values.
2.5 + 3.75
Line up the decimal points:
2.50
+ 3.75
---
6.25The bakery sold 6.25 pounds of cookies in total.
Adding Two Negative Rational Numbers
When both numbers are negative, add their absolute values and keep the negative sign.
Example: On Monday, the temperature dropped 3°F below the starting point.
On Tuesday, it dropped another 5°F from there.What is the total temperature change over the two days?
Solution:
Both changes are negative: -3 and -5.
Add the absolute values: 3 + 5 = 8
Keep the negative sign: -3 + (-5) = -8
The total temperature change is -8°F, meaning the temperature dropped 8 degrees.
Think of walking: if you walk 3 steps backward, then walk 5 more steps backward, you've moved a total of 8 steps backward from where you started.
Adding Rational Numbers with Different Signs
When you add two rational numbers with different signs-one positive and one negative-you need to find the difference between their absolute values. The result takes the sign of the number with the larger absolute value.
The Rule for Different Signs
To add a positive and a negative number:
- Find the absolute value of each number (ignore the signs temporarily)
- Subtract the smaller absolute value from the larger absolute value
- Give the result the sign of the number that had the larger absolute value
Example: A submarine is at a depth of -120 feet (120 feet below sea level).
It rises 45 feet.What is the submarine's new depth?
Solution:
Starting depth: -120 feet
Change: +45 feetWe need to calculate: -120 + 45
Absolute values: |-120| = 120 and |45| = 45
Subtract: 120 - 45 = 75
The larger absolute value (120) was negative, so the answer is negative: -75
The submarine's new depth is -75 feet, or 75 feet below sea level.
Example: Luis has $8.50.
He spends $12.00 on lunch.What is his account balance now?
Solution:
Starting amount: +8.50
Spending: -12.00We calculate: 8.50 + (-12.00)
Absolute values: |8.50| = 8.50 and |-12.00| = 12.00
Subtract: 12.00 - 8.50 = 3.50
The larger absolute value (12.00) was negative, so the answer is negative: -3.50
Luis's balance is -$3.50, meaning he owes $3.50.
Subtracting Rational Numbers
Subtraction of rational numbers is closely related to addition. In fact, every subtraction problem can be rewritten as an addition problem.
The Key Rule: Add the Opposite
To subtract a rational number, add its opposite. The opposite of a number has the same absolute value but the opposite sign.
\[ a - b = a + (-b) \]In this formula, \( a \) is the starting number, \( b \) is the number being subtracted, and \( -b \) is the opposite of \( b \).
The opposite of a number is also called its additive inverse. For example:
- The opposite of 7 is -7
- The opposite of -3 is 3
- The opposite of \( \frac{2}{5} \) is \( -\frac{2}{5} \)
- The opposite of -8.4 is 8.4
Subtracting a Positive Number
When you subtract a positive number, you add its opposite, which is negative.
Example: The temperature is 3°C.
It drops by 7°C.What is the new temperature?
Solution:
Starting temperature: 3°C
Drop means subtract: 3 - 7Rewrite as addition: 3 - 7 = 3 + (-7)
Find absolute values: |3| = 3 and |-7| = 7
Subtract: 7 - 3 = 4
The larger absolute value was negative, so: 3 + (-7) = -4
The new temperature is -4°C, or 4 degrees below zero.
Subtracting a Negative Number
When you subtract a negative number, you add its opposite, which is positive. This means subtracting a negative is the same as adding a positive.
Example: A diver is at -15 feet (15 feet below the surface).
The depth decreases by -8 feet.What is the diver's new depth?
Solution:
Starting depth: -15 feet
Change: subtract -8 means -15 - (-8)Rewrite: -15 - (-8) = -15 + 8
Absolute values: |-15| = 15 and |8| = 8
Subtract: 15 - 8 = 7
The larger absolute value was negative, so: -15 + 8 = -7
The diver's new depth is -7 feet, or 7 feet below the surface.
Think of it this way: if you owe someone $10 (-10) and they forgive $3 of that debt (subtract -3), you now owe only $7. Subtracting a negative improved your situation, just like adding a positive would.
Adding and Subtracting Fractions
When rational numbers are expressed as fractions, you need a common denominator before adding or subtracting. The same rules about signs still apply.
Same Denominator
When fractions have the same denominator, add or subtract the numerators and keep the denominator.
\[ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \]Example: Add \( \frac{5}{8} \) and \( \frac{3}{8} \).
What is the sum?
Solution:
Both fractions have denominator 8.
Add the numerators: 5 + 3 = 8
Keep the denominator: \( \frac{5}{8} + \frac{3}{8} = \frac{8}{8} = 1 \)
The sum is 1.
Different Denominators
When fractions have different denominators, find the least common denominator (LCD), convert each fraction, then add or subtract.
Example: Calculate \( \frac{2}{3} - \frac{1}{4} \).
What is the difference?
Solution:
Find the LCD of 3 and 4: LCD = 12
Convert each fraction:
\( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)
\( \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)Subtract the numerators: \( \frac{8}{12} - \frac{3}{12} = \frac{5}{12} \)
The difference is \( \frac{5}{12} \).
Fractions with Negative Signs
When fractions are negative, apply the same addition and subtraction rules for signed numbers.
Example: Add \( -\frac{3}{5} \) and \( -\frac{1}{2} \).
What is the sum?
Solution:
Both fractions are negative, so we add their absolute values and keep the negative sign.
Find the LCD of 5 and 2: LCD = 10
Convert each fraction:
\( \frac{3}{5} = \frac{6}{10} \)
\( \frac{1}{2} = \frac{5}{10} \)Add absolute values: \( \frac{6}{10} + \frac{5}{10} = \frac{11}{10} \)
Keep the negative sign: \( -\frac{3}{5} + \left(-\frac{1}{2}\right) = -\frac{11}{10} \) or \( -1\frac{1}{10} \)
The sum is \( -\frac{11}{10} \) or \( -1\frac{1}{10} \).
Adding and Subtracting Decimals
Decimal numbers are another form of rational numbers. Adding and subtracting decimals requires careful alignment and attention to signs.
Aligning Decimal Points
Always line up the decimal points vertically before adding or subtracting. This ensures that you're adding tenths to tenths, hundredths to hundredths, and so on.
Example: Calculate 12.75 - 8.3.
What is the difference?
Solution:
Write the numbers with aligned decimal points. Add a zero as a placeholder if needed.
12.75
- 8.30
---
4.45The difference is 4.45.
Decimals with Negative Signs
When decimals involve negative numbers, follow the addition and subtraction rules for signed numbers.
Example: A stock price changes by -2.35 points on Monday.
On Tuesday, it changes by +1.8 points.What is the total change over the two days?
Solution:
Total change = -2.35 + 1.8
Absolute values: |-2.35| = 2.35 and |1.8| = 1.8
Subtract: 2.35 - 1.80 = 0.55
The larger absolute value was negative, so the result is negative: -0.55
The total change is -0.55 points, meaning the stock dropped by 0.55 points overall.
Properties of Addition with Rational Numbers
Several important properties govern how addition works with rational numbers. These properties help you compute efficiently and understand the structure of mathematics.
Commutative Property of Addition
The commutative property states that the order of addition doesn't matter. For any rational numbers \( a \) and \( b \):
\[ a + b = b + a \]For example: \( 5 + (-3) = -3 + 5 = 2 \)
Associative Property of Addition
The associative property states that when adding three or more numbers, grouping doesn't matter. For any rational numbers \( a \), \( b \), and \( c \):
\[ (a + b) + c = a + (b + c) \]For example: \( (2 + 4) + 6 = 2 + (4 + 6) = 12 \)
Identity Property of Addition
The identity property states that adding zero to any number gives that same number. Zero is called the additive identity.
\[ a + 0 = a \]For example: \( -7 + 0 = -7 \)
Inverse Property of Addition
The inverse property states that every number has an opposite (additive inverse) such that their sum is zero.
\[ a + (-a) = 0 \]For example: \( \frac{3}{4} + \left(-\frac{3}{4}\right) = 0 \)
Multi-Step Problems with Rational Numbers
Real-world situations often require multiple additions and subtractions. Breaking the problem into clear steps makes complex calculations manageable.
Example: Maya starts with $45.00 in her account.
She deposits $25.50.
She then writes a check for $38.75.
Finally, she withdraws $15.00.What is her final balance?
Solution:
Starting balance: $45.00
After deposit: 45.00 + 25.50 = $70.50
After check: 70.50 - 38.75 = $31.75
After withdrawal: 31.75 - 15.00 = $16.75
Maya's final balance is $16.75.
Example: A hiker starts at an elevation of 1,200 feet.
She climbs up 450 feet.
Then she descends 600 feet.
Finally, she climbs up another 275 feet.What is her final elevation?
Solution:
Starting elevation: 1,200 feet
After first climb: 1,200 + 450 = 1,650 feet
After descent (subtract): 1,650 - 600 = 1,050 feet
After second climb: 1,050 + 275 = 1,325 feet
The hiker's final elevation is 1,325 feet.
Using Number Lines
A number line is a visual tool that helps you see addition and subtraction of rational numbers. Positive numbers are to the right of zero, and negative numbers are to the left.
To add a positive number, move to the right. To add a negative number (or subtract a positive), move to the left.
Think of the number line as a horizontal thermometer or a timeline. Moving right means increasing; moving left means decreasing.
For example, to compute \( -2 + 5 \):
- Start at -2 on the number line
- Move 5 units to the right
- You land on 3
To compute \( 4 - 7 \), which is the same as \( 4 + (-7) \):
- Start at 4 on the number line
- Move 7 units to the left
- You land on -3
Common Mistakes to Avoid
When working with rational numbers, certain errors occur frequently. Being aware of them helps you check your work.
Sign Errors
Forgetting to apply the correct rule for different signs is a common mistake. Remember: when adding numbers with different signs, subtract absolute values and use the sign of the number with the larger absolute value.
Double Negative Confusion
Subtracting a negative number means adding a positive. Many students forget this rule. Always rewrite subtraction as "add the opposite" to avoid confusion.
Decimal Misalignment
When adding or subtracting decimals, failing to line up decimal points leads to incorrect place values. Always write numbers vertically with aligned decimal points.
Fraction Denominator Errors
When adding fractions with different denominators, you must find a common denominator first. Never add denominators directly.
Practical Applications
Addition and subtraction of rational numbers appear throughout daily life and various fields.
Finance
Bank accounts use positive numbers for deposits and negative numbers for withdrawals or debts. Calculating balances requires adding and subtracting rational numbers, including decimals.
Temperature
Temperature changes involve both positive and negative values. If the temperature starts at -5°F and rises by 12°F, you calculate -5 + 12 = 7°F.
Elevation
Elevations below sea level are negative, while those above are positive. Finding total elevation change requires subtracting starting elevation from ending elevation.
Sports Statistics
In golf, scores below par are negative and scores above par are positive. Finding a player's total score requires adding positive and negative integers.
Understanding how to add and subtract rational numbers with confidence gives you powerful tools for solving problems in mathematics and in the world around you. Whether you're working with money, measurements, or abstract numbers, the principles remain the same: pay attention to signs, follow the rules for same and different signs, and always double-check your work.
