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Number Bases · Form 4

Number Bases: Worked Examples (Medium)

Two-to-three-step number base work: converting base ten into another base, and adding or subtracting within a base with carrying and borrowing. For students ready to handle the reverse direction and column arithmetic.

Written by the spmmath.com.my editorial team.

Worked example 1

Convert 92₁₀ to base 5.

  1. Divide repeatedly by 5 and record each remainder: 92 ÷ 5 = 18 remainder 2.
  2. 18 ÷ 5 = 3 remainder 3.
  3. 3 ÷ 5 = 0 remainder 3.
  4. Read the remainders from bottom to top: 3, 3, 2.
  5. Check: (3×25) + (3×5) + (2×1) = 75 + 15 + 2 = 92. ✓

Worked example 2

Find 1011₂ + 110₂, giving the answer in base 2.

  1. Align the numbers by their right-hand (units) column and add column by column, carrying when a column reaches 2.
  2. Units: 1 + 0 = 1, write 1.
  3. Twos: 1 + 1 = 2, write 0 carry 1.
  4. Fours: 0 + 1 + 1(carry) = 2, write 0 carry 1.
  5. Eights: 1 + 0 + 1(carry) = 2, write 0 carry 1; the final carry gives a leading 1.
  6. Reading the columns: 1 0 0 0 1.
  7. Check in base ten: 11 + 6 = 17, and 10001₂ = 16 + 1 = 17. ✓

Worked example 3

Find 623₈ − 157₈, giving the answer in base 8.

  1. Align by the units column and subtract column by column; when you borrow, you borrow a group of 8.
  2. Units: 3 − 7 needs a borrow. Take 1 from the eights column: (3 + 8) − 7 = 11 − 7 = 4.
  3. Eights: the 2 became 1 after lending. 1 − 5 needs a borrow: (1 + 8) − 5 = 9 − 5 = 4.
  4. Sixty-fours: the 6 became 5 after lending. 5 − 1 = 4.
  5. Reading the columns: 4 4 4.
  6. Check in base ten: 403 − 111 = 292, and 444₈ = (4×64)+(4×8)+4 = 256+32+4 = 292. ✓

Worked example 4

Convert 200₁₀ to base 2.

  1. Divide repeatedly by 2 and record each remainder:
  2. 200÷2 = 100 r 0, 100÷2 = 50 r 0, 50÷2 = 25 r 0, 25÷2 = 12 r 1.
  3. 12÷2 = 6 r 0, 6÷2 = 3 r 0, 3÷2 = 1 r 1, 1÷2 = 0 r 1.
  4. Read the remainders from the last to the first: 11001000.

Worked example 5

Find 234₅ + 143₅, giving your answer in base 5.

  1. Add column by column in base 5, carrying when a column reaches 5.
  2. Units: 4 + 3 = 7 = 1×5 + 2, write 2 carry 1.
  3. Fives: 3 + 4 + 1(carry) = 8 = 1×5 + 3, write 3 carry 1.
  4. Twenty-fives: 2 + 1 + 1(carry) = 4, write 4.
  5. The result is 432₅.

Worked example 6

Convert 1011010₂ to base 8.

  1. Group the binary digits in threes from the right: 1 011 010.
  2. Pad the leftmost group with zeros: 001 011 010.
  3. Convert each group of three: 001 = 1, 011 = 3, 010 = 2.
  4. Write the groups together: 132.

Worked example 7

Convert 150₁₀ to base 8.

  1. Divide repeatedly by 8, recording the remainders: 150 ÷ 8 = 18 remainder 6
  2. 18 ÷ 8 = 2 remainder 2
  3. 2 ÷ 8 = 0 remainder 2
  4. Read the remainders from bottom to top: 2, 2, 6

Worked example 8

Find 342₅ − 124₅, giving your answer in base 5.

  1. Units digit: 2 − 4 cannot be done directly, so borrow 1 group of five from the tens digit: (2 + 5) − 4 = 3; the tens digit reduces from 4 to 3.
  2. Tens digit: 3 − 2 = 1
  3. Hundreds digit: 3 − 1 = 2
  4. Combine the digits: 213

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

How do I add or subtract numbers directly in a non-decimal base?

Work column by column like normal addition or subtraction, but carry or borrow according to that base's value instead of ten, for example, in base two a carry happens once a column reaches 2. Practising a few by hand builds the habit better than converting to base ten every time.

What's the best way to convert between two bases that aren't base ten?

Convert the number to base ten first, then convert from base ten to the target base, trying to jump directly between two unfamiliar bases usually causes errors. Keep the two conversion steps separate and check your working at each stage.

Why do I lose marks even when my final answer in a different base is correct?

Examiners award method marks for each conversion step, such as the division-remainder table or the place-value expansion. Skipping straight to the answer without showing this working means you lose marks even if the final digits are correct.

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