Rounding off the digits: what to do with a 5

Keep the digits you can trust

A calculator gives more digits than a measurement can support. Round off to the number of significant figures you need, always to the nearest value.

Look at the first digit you drop. More than 5: raise the last kept digit by one. Less than 5: leave it as it is. So 2.512 × 10³ m becomes 2.5 × 10³ m.

Exactly 5 is the tie. Your book uses an arbitrary rule: if the digit before the 5 is odd, add one; if it is even, leave it. So 4.45 × 10² m → 4.4 × 10² m, but 4.55 × 10² m → 4.6 × 10² m.

If digits follow the 5, it is no longer a tie. 4.452 × 10² m is closer to 4.5 × 10² m than to 4.4 × 10² m, so it rounds up.

Dropped digit above 5: round up. Below 5: keep. Exactly 5: odd before it rounds up, even before it stays.

Key terms

Rounding off

Cutting a number to the significant figures you need.

Retained digit

The digit that may go up by one.

Odd/even rule

Odd before the 5: add one. Even before it: no change.

Logic rule

When digits follow the 5, go to whichever value is nearer.

Short questions with model answers

1. Q1. Round 3.4567 × 10⁴ kg to 3 significant figures.

   • 3.45 | 67
   • 3.45 → 3.46

   3.46 × 10⁴ kg.

2. Q2. Round 4.452 × 10² m to 2 significant figures. Why is it not 4.4 × 10² m?

   • 4.4 | 52
   • 4.452 − 4.4 = 0.052 ; 4.5 − 4.452 = 0.048

   4.5 × 10² m.

Common mistakes

• ✗ Always rounding an exact 5 up: 4.45 × 10² m → 4.5 × 10² m.

  ✓ Your book: the digit before the 5 is even (4), so it stays: 4.4 × 10² m.

• ✗ Using the odd/even rule on 4.452 × 10² m.

  ✓ Digits after the 5 break the tie; go to the nearer value, 4.5 × 10² m.

• ✗ Writing 2.512 × 10³ m rounded as just 2.5 m.

  ✓ Rounding changes the digits, not the size: 2.5 × 10³ m.

MCQs

1. 1. Round 2.512 × 10³ m to 2 significant figures:

   1. (a) 2.6 × 10³ m
   2. (b) 2.5 × 10³ m
   3. (c) 2.51 × 10³ m
   4. (d) 2.5 m
   Show answer

   (b) The dropped 1 is less than 5.

2. 2. Round 3.4567 × 10⁴ kg to 3 significant figures:

   1. (a) 3.45 × 10⁴ kg
   2. (b) 3.457 × 10⁴ kg
   3. (c) 3.5 × 10⁴ kg
   4. (d) 3.46 × 10⁴ kg
   Show answer

   (d) The dropped 6 is more than 5.

3. 3. By your book's rule, 4.55 × 10² m to 2 significant figures is:

   1. (a) 4.6 × 10² m
   2. (b) 4.5 × 10² m
   3. (c) 4.55 × 10² m
   4. (d) 4.0 × 10² m
   Show answer

   (a) The digit before the 5 is odd (5), so add one.

4. 4. Round 4.452 × 10² m to 2 significant figures:

   1. (a) 4.4 × 10² m
   2. (b) 4.45 × 10² m
   3. (c) 4.5 × 10² m
   4. (d) 4.0 × 10² m
   Show answer

   (c) It is closer to 4.5 than to 4.4 (p. 23).

5. 5. Under your book's rule, when the dropped digit is exactly 5 and the digit before it is even:

   1. (a) Add one to it
   2. (b) Leave it as it is
   3. (c) Take one away
   4. (d) Keep the 5 as well
   Show answer

   (b) Even before the 5: it remains the same (p. 22).

Quick revision

• Round to the nearest value: above 5 up, below 5 unchanged.
• An exact 5: odd before it goes up, even before it stays (the book's rule).
• Digits after the 5 break the tie: go to the nearer value.