Rounding Rules
Rounding makes numbers simpler while keeping them close to the original value. The most common rule is "round half up" — if the dropped digit is 5 or greater, round up.
Standard Rounding Rule:
If next digit ≥ 5 → round up
If next digit < 5 → round down
3.146 rounded to 2 decimal places = 3.15
3.144 rounded to 2 decimal places = 3.14
If next digit ≥ 5 → round up
If next digit < 5 → round down
3.146 rounded to 2 decimal places = 3.15
3.144 rounded to 2 decimal places = 3.14
Common Rounding Applications
- Currency: round prices to 2 decimal places ($3.456 → $3.46)
- Measurements: round to significant figures for accuracy
- Statistics: report results to meaningful decimal places
Rounding Rules — All Methods
| Method | Rule | 3.145 → 2dp | 3.155 → 2dp |
|---|---|---|---|
| Round Half Up (standard) | ≥0.5 rounds up | 3.15 | 3.16 |
| Round Half Down | >0.5 rounds up | 3.14 | 3.16 |
| Round Half to Even (Banker's) | 0.5 rounds to nearest even | 3.14 | 3.16 |
| Truncate (floor) | Always round toward zero | 3.14 | 3.15 |
Significant Figures vs Decimal Places
Decimal places: count digits AFTER decimal point
0.00456 to 2dp = 0.00 (meaningless!)
Significant figures: count ALL non-zero digits and zeros between/after them
0.00456 to 2 sig.figs = 0.0046
12,345 to 3 sig.figs = 12,300
0.00456 to 2dp = 0.00 (meaningless!)
Significant figures: count ALL non-zero digits and zeros between/after them
0.00456 to 2 sig.figs = 0.0046
12,345 to 3 sig.figs = 12,300
Rounding to Nearest 10, 100, 1000
Round 7,483 to nearest 100:
Look at tens digit: 8 ≥ 5, so round up
7,483 → 7,500
Round 7,449 to nearest 100:
Look at tens digit: 4 < 5, so round down
7,449 → 7,400
Look at tens digit: 8 ≥ 5, so round up
7,483 → 7,500
Round 7,449 to nearest 100:
Look at tens digit: 4 < 5, so round down
7,449 → 7,400
Real-World Applications
- Currency: Prices rounded to 2 decimal places ($3.456 → $3.46)
- Science: Measurement results reported to appropriate significant figures
- Statistics: Final results rounded to meaningful precision (percentage to 1dp)
- Engineering: Tolerances — parts may be specified to 3 decimal places
- Everyday life: Rounding grocery totals, time estimates, distances
Rounding in Financial Calculations
Rounding has important implications in finance. Banks and financial systems use specific rounding rules to ensure consistency and fairness:
- Banker's Rounding (Round Half to Even): When the dropped digit is exactly 5, round to the nearest even number. This prevents systematic upward bias over many transactions. Used in accounting, statistics, and IEEE 754 floating-point arithmetic.
- Currency rounding: Most currencies round to 2 decimal places. Some (Japanese yen, Korean won) use 0 decimal places.
- Tax calculations: Different tax authorities have specific rounding rules — some round down (floor), others use standard rounding.
Rounding Errors — Why They Matter
Adding 1000 items each rounded from 0.5000 to 0.50:
With round-half-up: 1000 × $0.50 = $500.00 (exact ✓)
But: 1000 items at true $0.505 = $505.00
Rounding error = $5.00 — not trivial in finance!
With round-half-up: 1000 × $0.50 = $500.00 (exact ✓)
But: 1000 items at true $0.505 = $505.00
Rounding error = $5.00 — not trivial in finance!
This is why large-scale financial systems use arbitrary-precision arithmetic and carefully defined rounding modes rather than simple float arithmetic.