Base Systems • Topic 3 of 3

Base Arithmetic

You can add, subtract and multiply directly in any base — the only change is the carry/borrow threshold. When a column total reaches the base b, you write the remainder and carry 1 (you carry at b, not at 10). In base 5, 3 + 4 = 7 = 1·5 + 2, so you write 2 and carry 1. Subtraction works the same in reverse: when you borrow, you add b to the current column, not 10. In base 8, borrowing turns a 3 into 3 + 8 = 11 before subtracting. Multiplication is ordinary long multiplication, but every partial product is reduced using the base: in base 6, 4 × 5 = 20 decimal = 3·6 + 2, written 32 in base 6. Two tricks save time in CAT: (1) for a quick check, convert each operand to decimal, do the sum, and convert back; (2) for clean values, spot that adding 1 to the largest n-digit number rolls over to a power of the base (in base 7, 666 + 1 = 1000).

✅ Solved examples

1. Add the base-5 numbers 34 and 23.
Units: 4 + 3 = 7 = 1·5 + 2, write 2 carry 1. Next: 3 + 2 + 1 = 6 = 1·5 + 1, write 1 carry 1. Answer 112 in base 5. (Check: 19 + 13 = 32 = 112_5.)
2. Subtract 17 from 43 in base 8.
Units: 3 − 7 needs a borrow: 3 + 8 = 11, 11 − 7 = 4, write 4. Tens: 4 became 3 after the borrow, 3 − 1 = 2. Answer 24 in base 8. (Check: 35 − 15 = 20 = 24_8.)
3. Multiply 4 by 5 in base 6.
4 × 5 = 20 in decimal = 3·6 + 2, so the result is 32 in base 6.
4. In base 7, what is 666 + 1?
666 is the largest 3-digit base-7 number. Adding 1 rolls every digit over: 666 + 1 = 1000 in base 7 (= 343 in decimal).

✏️ Practice — try these, take hints as needed

1. Add 23 and 14 in base 5.
Units: 3 + 4 = 7.
7 = 1·5 + 2, so write 2 carry 1.
Tens: 2 + 1 + carry 1.
42 (base 5)
2. Add 27 and 35 in base 8.
Units: 7 + 5 = 12 decimal.
12 = 1·8 + 4, write 4 carry 1.
Tens: 2 + 3 + 1.
64 (base 8)
3. Subtract 25 from 52 in base 6.
Units: 2 − 5 needs a borrow.
2 + 6 = 8, then 8 − 5 = 3.
Tens: 5 became 4, then 4 − 2.
23 (base 6)
4. Multiply 3 by 4 in base 5.
3 × 4 = 12 in decimal.
12 = 2·5 + 2.
Write the remainder, carry the quotient.
22 (base 5)
5. In base 4, compute 333 + 1.
333 is the largest 3-digit base-4 number.
Adding 1 rolls every digit over.
It becomes a 4-digit power of the base.
1000 (base 4)

📝 Topic test — 8 questions

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