Operations in Number Bases · Hook

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Ever wondered what 'carry one' means when you count in fives?

Zainab sells oranges in heaps of 5, and 5 heaps make a basket. When she adds two sales together, every 5 loose oranges become 1 heap — that is carrying in base 5! The same idea works for subtraction and multiplication.

Revision notes

📒 Operations in Number Bases: lesson notes

Add, subtract and multiply numbers in base 2 and base 5 — carrying and borrowing in the base, not in ten.

Zainab sells oranges in heaps of 5, and 5 heaps make a basket. When she adds two sales together, every 5 loose oranges become 1 heap — that is carrying in base 5! The same idea works for subtraction and multiplication.

Adding: carry the base, not ten

Add column by column from the right. When a column adds up to the base or more, write what is left over and carry 1 for every group of the base.

345 + 235: units 4 + 3 = 7 = 1 × 5 + 2, so write 2, carry 1. Next: 3 + 2 + 1 = 6 = 1 × 5 + 1, so write 1, carry 1. Answer: 1125.

Check: 345 = 19 and 235 = 13; 19 + 13 = 32, and 1125 = 25 + 5 + 2 = 32 ✓.

🔑 Key idea: In base b, every b in a column carries 1 to the next column.

Subtracting: borrowing gives you the base

When the top digit is too small, borrow 1 from the next column. In base 5 that borrowed 1 is worth 5, not 10.

425 − 145: units 2 is less than 4, so borrow: 2 + 5 − 4 = 3. Next column: 4 − 1 (lent) − 1 = 2. Answer: 235.

Check: 22 − 9 = 13, and 235 = 2 × 5 + 3 = 13 ✓. In base 2, a borrowed 1 is worth 2.

🔑 Key idea: Borrowing 1 from the next column gives you b in base b.

Multiplying: multiply, then carry in the base

Multiply each digit, starting from the right, and carry groups of the base. 235 × 3: units 3 × 3 = 9 = 1 × 5 + 4, write 4, carry 1. Next: 2 × 3 + 1 = 7 = 1 × 5 + 2, write 2, carry 1. Answer: 1245.

In base 2, multiplying is easy: × 1 copies the number and × 102 shifts it one place left. Add the shifted copies in base 2.

If you are stuck, change to base ten, work it out, and change back — it is also the best way to check.

🔑 Key idea: Work in the base all the way through; check using base ten.

Worked example

Work out 4305 + 4115. Type your answer in base 5 digits.

  1. Step 1: Add column by column, from the right

    Units column: 0 + 1 = 1 → write 1

    Column 2: 3 + 1 = 4 → write 4

    Column 3: 4 + 4 = 8 = 1 × 5 + 3 → write 3, carry 1

    The last carry, 1, goes at the front.

  2. Step 2: Answer

    4305 + 4115 = 13415

  3. Step 3: Check in base ten

    4305 = 115 and 4115 = 106

    115 + 106 = 221, and 13415 = 221 ✓

Answer: 1,341

Summary

  • Base 2 uses the digits 0 and 1; base 5 uses 0 to 4.
  • Adding: when a column reaches the base, write the remainder and carry 1.
  • Subtracting: a borrowed 1 is worth the base (5 in base 5, 2 in base 2).
  • Multiplying: multiply each digit and carry groups of the base.
  • Check any answer by converting to base ten.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.