🎬 Regroup within the chosen base
In base two, two units regroup as one group of two.
🎯 By the end of this lesson...
You should be able to:
- Add numbers written in the same base.
- Explain when carrying is required.
- Carry correctly in bases other than Base 10.
- Subtract numbers written in the same base.
- Borrow correctly in bases other than Base 10.
- Check calculations by converting to Base 10.
- Solve simple equations involving number bases.
- Identify invalid answers in a stated base.
- Apply number-base operations to practical situations.
🤔 Think About This...
How does a computer add?
In ordinary Base 10:
But Base 2 does not contain the digit 2.
Therefore, in Base 2:
The answer looks different, but it still represents a value of two in Base 10.
1️⃣ Revisiting the Meaning of the Base
The base tells us how many units make one unit in the next place-value column.
Base 2
Base 5
Base 8
💡 Did You Know?
Binary addition is used inside computers whenever they perform calculations.
The last result means write 0 in the current column and carry 1 into the next column.
2️⃣ Adding Without Carrying
If each column total is smaller than the base, addition works just like ordinary column addition.
Example 1: Calculate 102₃ + 120₃
Base 3 allows only the digits 0, 1 and 2.
Work from right to left:
- 2 + 0 = 2
- 0 + 2 = 2
- 1 + 1 = 2
3️⃣ Adding with Carrying
Carry whenever a column total is equal to or greater than the base.
Example 2: Calculate 24₅ + 13₅
Begin with the units column.
But Base 5 cannot contain the digit 7.
Rewrite 7 as:
Write 2 and carry 1.
In the next column:
🧠 StudyNest Method: Adding in Another Base
Step 2: Write the numbers vertically, lining up place values.
Step 3: Add from right to left.
Step 4: If a column total is smaller than the base, write it directly.
Step 5: If the total reaches or exceeds the base, divide it into groups of the base.
Step 6: Write the remainder and carry the number of complete groups.
Step 7: Check that every digit in the answer is smaller than the base.
4️⃣ Binary Addition
Example 3: Calculate 1011₂ + 110₂
Align the place values.
Work carefully from the right:
- 1 + 0 = 1
- 1 + 1 = 10₂. Write 0 and carry 1.
- 0 + 1 + 1 carried = 10₂. Write 0 and carry 1.
- 1 + 0 + 1 carried = 10₂. Write 0 and carry 1.
- Write the final carried 1.
Check in Base 10
5️⃣ Addition in Base 8
Example 4: Calculate 357₈ + 246₈
Add the units:
In Base 8:
Write 5 and carry 1.
In Base 8:
Write 2 and carry 1.
6️⃣ Subtracting Without Borrowing
If the top digit in every column is large enough, subtract normally.
Example 5: Calculate 432₅ − 121₅
- 2 − 1 = 1
- 3 − 2 = 1
- 4 − 1 = 3
7️⃣ Borrowing in Another Base
When borrowing, one unit from the next column is worth the value of the base in the current column.
Base 10
Borrowing 1 ten gives:
Base 5
Borrowing 1 five gives:
Base 2
Borrowing 1 two gives:
8️⃣ Subtracting with Borrowing
Example 6: Calculate 42₅ − 14₅
In the units column, 2 − 4 cannot be done without borrowing.
Borrow 1 from the fives column.
The 4 in the fives column becomes 3.
🧠 StudyNest Method: Subtracting in Another Base
Step 2: Line up matching place values.
Step 3: Subtract from right to left.
Step 4: If the top digit is large enough, subtract normally.
Step 5: If the top digit is too small, borrow 1 from the next column.
Step 6: Add the value of the base to the current top digit.
Step 7: Reduce the digit from which you borrowed by 1.
Step 8: Check that every final digit is valid in the base.
9️⃣ Binary Subtraction
Example 7: Calculate 1010₂ − 11₂
Align the place values:
In the units column, 0 − 1 requires borrowing.
In Base 2, borrowing 1 gives 2 units.
Continue through the remaining columns.
Check in Base 10
🔟 Checking Calculations Using Base 10
A useful way to verify a difficult number-base calculation is to convert all values to Base 10.
Example 8: Check 24₅ + 13₅ = 42₅
1️⃣1️⃣ Adding Numbers Written in Different Bases
Numbers must be converted to the same base before they can be added directly.
Example 9: Calculate 101₂ + 12₃
Convert both numbers to Base 10.
1️⃣2️⃣ Simple Equations Involving Number Bases
A letter may be used to represent an unknown base.
Example 10: Find b if 23b = 17₁₀
Expand the place values.
Therefore:
1️⃣3️⃣ Equations with a Missing Digit
Example 11: Find x if 2x₅ = 13₁₀
Expand the Base 5 number.
🧠 StudyNest Method: Solving a Base Equation
Step 2: Expand the number using place value.
Step 3: Form an ordinary algebraic equation.
Step 4: Solve the equation.
Step 5: Check that the answer is valid.
A base must be greater than every digit in the number.
A missing digit must be smaller than the base.
🌍 Real-Life Example: Binary Counters
Example 12
A digital counter displays:
It then increases by:
Add the binary values:
In Base 10:
🎯 Exam Tips
⚠ Common Mistakes
Carrying in groups of 10 regardless of the stated base.
Writing a digit in the answer that is equal to or greater than the base.
Forgetting to include the carried digit in the next column.
Borrowing 10 units instead of borrowing the value of the base.
Adding numbers written in different bases without first converting them to a common base.
Solving an equation correctly but accepting an invalid base or digit.
🎮 Your Turn!
🟢 Foundation Questions
-
Calculate:
12₃ + 10₃ -
Calculate:
21₄ + 12₄ -
Calculate:
23₅ + 11₅ -
Calculate:
101₂ + 10₂ -
Calculate:
43₅ − 21₅ -
Calculate:
322₄ − 101₄
🔵 Developing Questions
-
Calculate:
24₅ + 32₅ -
Calculate:
346₇ + 125₇ -
Calculate:
111₂ + 101₂ -
Calculate:
1011₂ + 111₂ -
Calculate:
42₅ − 13₅ -
Calculate:
201₃ − 22₃ -
Calculate:
1100₂ − 101₂ - Check your answer to Question 8 by converting both sides to Base 10.
🔴 Challenge Questions
-
Calculate:
657₈ + 276₈ -
Calculate:
10000₂ − 111₂ -
Calculate and give your answer in Base 10:
101₂ + 12₃ -
Find b:
24b = 18₁₀ -
Find b:
31b = 16₁₀ -
Find x:
3x₅ = 18₁₀ -
Find x:
1x2₄ = 26₁₀ - A binary counter displays 1101₂ and increases by 101₂. What is the new binary display?
✅ Show Answers
- 12₃ + 10₃ = 22₃
- 21₄ + 12₄ = 33₄
- 23₅ + 11₅ = 34₅
- 101₂ + 10₂ = 111₂
- 43₅ − 21₅ = 22₅
- 322₄ − 101₄ = 221₄
-
24₅ + 32₅
Units: 4 + 2 = 6 = 11₅
Write 1 and carry 1.
Fives: 2 + 3 + 1 = 6 = 11₅
Answer = 111₅ -
346₇ + 125₇
= 504₇ - 111₂ + 101₂ = 1100₂
- 1011₂ + 111₂ = 10010₂
- 42₅ − 13₅ = 24₅
- 201₃ − 22₃ = 102₃
- 1100₂ − 101₂ = 111₂
-
346₇ = 3 × 49 + 4 × 7 + 6
= 181₁₀
125₇ = 1 × 49 + 2 × 7 + 5
= 68₁₀
181 + 68 = 249
504₇ = 5 × 49 + 4
= 249₁₀ - 657₈ + 276₈ = 1155₈
- 10000₂ − 111₂ = 1001₂
-
101₂ = 5₁₀
12₃ = 5₁₀
Total = 10₁₀ -
24b = 2b + 4
2b + 4 = 18
2b = 14
b = 7 -
31b = 3b + 1
3b + 1 = 16
3b = 15
b = 5 -
3x₅ = 3 × 5 + x
15 + x = 18
x = 3 -
1x2₄ = 1 × 16 + x × 4 + 2
18 + 4x = 26
4x = 8
x = 2 -
1101₂ + 101₂
= 10010₂
🧠 Remember This
Carry when a column total reaches the value of the base.
When borrowing, one borrowed unit is worth the value of the base.
Every digit in the answer must be smaller than the base.
Convert to Base 10 when checking difficult calculations.
Base equations can be rewritten as ordinary algebraic equations.
Always check that a missing base or digit is valid.
🎯 Before Moving On...
- ✅ Can I add numbers written in the same base?
- ✅ Do I know when to carry?
- ✅ Can I carry correctly in Base 2, Base 5 and Base 8?
- ✅ Can I subtract numbers written in the same base?
- ✅ Can I borrow the correct value?
- ✅ Can I check a calculation using Base 10?
- ✅ Can I solve for an unknown base?
- ✅ Can I solve for a missing digit?
- ✅ Can I check whether my answer is valid?
🎉 Well Done!
You can now perform calculations in different number bases.
Check the base → Line up place values → Carry or borrow using the base → Check every digit → Verify if needed
📈 Your Progress
Lesson 13 of 19