Number Bases · 2.1.3
Adding and subtracting in bases
Students perform addition and subtraction directly in a given base using column arithmetic: when a column's sum reaches the value of the base, they carry 1 to the next column; when subtracting and a digit is too small, they borrow an amount equal to the base from the next column.
The official learning standard (2.1.3)
“Perform calculations involving addition and subtraction of numbers in various bases.”
What it means
Students perform addition and subtraction directly in a given base using column arithmetic: when a column's sum reaches the value of the base, they carry 1 to the next column; when subtracting and a digit is too small, they borrow an amount equal to the base from the next column.
How it is examined
This is examined in Paper 1 as a short calculation item, and sometimes in Paper 2 combined with base conversion or embedded in a larger number-base problem that requires an addition or subtraction step in a non-decimal base.
Worked example
Calculate 1011₂ + 110₂. Give your answer in base 2.
- Align the numbers by place value: 1011 and 0110.
- Add the units column: 1 + 0 = 1.
- Add the twos column: 1 + 1 = 10₂, write 0, carry 1.
- Add the fours column: 0 + 1 + carry 1 = 10₂, write 0, carry 1.
- Add the eights column: 1 + 0 + carry 1 = 10₂, write 0, carry 1.
- Write down the final carry: 1.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Do I have to convert to base 10 first to add or subtract?
No, you can add or subtract directly in the given base using column arithmetic, as long as you carry or borrow using the base's value instead of 10. Converting to base 10 is optional and only useful for checking your answer.
How does borrowing work in subtraction in base 2?
If the top digit is smaller than the bottom digit in a column, borrow 1 from the next column to the left, that borrowed 1 is worth 2 in base 2 (or the base's value generally), added to the top digit before subtracting.
How can I check my answer is correct?
Convert both original numbers and your final answer into base 10 using place values, then check that the base 10 addition or subtraction gives the same result as your base 10 answer.