Rounding is one of the most practical mathematical skills. In everyday life, we round constantly: a supermarket price of £3.9971 becomes £3.99, a distance of 3.7 miles gets called "about 4 miles", and a room measuring 3.847 metres is noted as 3.85 m. Rounding makes numbers manageable without sacrificing meaningful precision.
In science, rounding is not just convenient but essential. Measurement precision is limited by instruments: a ruler accurate to 1 mm cannot meaningfully record 13.4871 cm. Reporting more digits than your instrument can reliably produce is scientifically misleading. The concept of significant figures captures this: the number of meaningful digits a measurement contains, regardless of decimal position.
In finance, rounding is legally and operationally significant. Tax calculations, VAT, currency conversion, and interest all require defined rounding rules to ensure consistency across millions of transactions.
The standard half-up rounding rule used in UK schools (GCSE AQA, Edexcel, OCR mark schemes) is:
Example — round 3.14159 to 2 decimal places: the second decimal digit is 4; the third decimal digit is 1 (which is less than 5), so we round down, giving 3.14.
Example — round 12.685 to 2dp: the second digit is 8; the third is 5 (5 or more), so round up: 8 + 1 = 9, giving 12.69.
Significant figures (sig figs) count from the first non-zero digit. To round 0.0004567 to 2 significant figures: the first significant digit is 4 (position 4 after decimal); the second is 5; look at the next digit, 6: round up the 5 to 6; result is 0.0046. See our dedicated Significant Figures Calculator for full rules and examples.
The standard "round half up" rule has a systematic bias: whenever a number ends in exactly 5, it always rounds upward. Over millions of transactions, this creates a small but measurable bias in the upward direction. Banker's rounding eliminates this bias by rounding to the nearest even number when the digit is exactly 5.
Banker's rounding is used in IEEE 754 floating-point arithmetic (the standard underpinning almost all computer calculations), Python 3's built-in round() function, Microsoft Excel's ROUND function (actually uses half-up), and many banking systems. When half-up and banker's rounding give different results for the same input, it only happens when the digit to round is exactly 5 with no further digits — which is common in test problems but relatively rare in real-world continuous data.
Rounding errors have caused some famous real-world failures. The Patriot missile failure in 1991 (Gulf War) resulted from a time-rounding error: the system clock's time was stored as a 24-bit integer representing tenths of a second. After 100 hours of continuous operation, the accumulated rounding error of 0.0000000951 seconds per tick grew to 0.34 seconds — enough to miscalculate a missile's position by 687 metres, causing the system to miss an incoming Scud missile, which killed 28 soldiers.
In financial computing, the EU Prospectus Directive mandates specific rounding rules for fund return calculations to prevent misleading investors. The UK Financial Conduct Authority (FCA) requires that consumer credit charges be rounded to the nearest penny in specific ways defined in the Consumer Credit Act.
Rounding to the nearest 10: look at the units digit. If 5 or more, round up; if 4 or less, round down. Example: 347 rounds to 350 (units digit = 7, round up). 342 rounds to 340 (units digit = 2, round down).
Rounding to the nearest 100: look at the tens digit. Example: 1,482 rounds to 1,500 (tens digit = 8, round up). 1,437 rounds to 1,400 (tens digit = 3, round down).
Rounding to the nearest 1,000: look at the hundreds digit. Example: 7,623 rounds to 8,000 (hundreds digit = 6, round up). 7,423 rounds to 7,000 (hundreds digit = 4, round down).
A useful trick: to round to the nearest N, divide by N, round to the nearest integer, then multiply back by N. For example, to round 73 to the nearest 25: 73/25 = 2.92 → round to 3 → 3 × 25 = 75.
Truncation simply removes digits beyond a specified position without any rounding adjustment. It always moves the value toward zero. For example:
Truncation is common in computing (integer casting, bit shifting) and in some financial contexts where rounding up would overcharge customers (e.g. some VAT calculations). It introduces a systematic downward bias and is less statistically fair than rounding.
Excel provides several rounding functions for different needs:
=ROUND(number, digits) — Standard half-up rounding to the specified decimal places. Negative digits round to tens, hundreds, etc.: ROUND(1234, -2) = 1200.=ROUNDUP(number, digits) — Always rounds away from zero, regardless of the digit following.=ROUNDDOWN(number, digits) — Always rounds toward zero (truncates).=MROUND(number, multiple) — Rounds to the nearest multiple of any number. MROUND(17, 5) = 15.=CEILING(number, significance) — Always rounds up to the next multiple.=FLOOR(number, significance) — Always rounds down to the previous multiple.Note: Excel's ROUND function uses half-up (not banker's rounding), even for exactly 0.5. This differs from Python's built-in round(), which uses banker's rounding.
In the UK (and most countries), monetary amounts are always expressed to exactly 2 decimal places, representing pounds and pence. This means 2dp is the standard for all sterling calculations. When a calculation results in a third or further decimal place (common in percentage calculations), the standard practice is to round to 2dp using half-up rounding. For example, VAT at 20% on a £7.99 item: £7.99 × 0.20 = £1.598 → rounded to £1.60.
Currency conversion often produces very long decimal sequences; banks typically round to 4 decimal places for exchange rates but deliver transactions rounded to 2dp.
When a measurement or number is given to a specific degree of accuracy, it could represent any value within a range. The lower bound is the smallest value that would round to the given number; the upper bound is the smallest value that would round to the next number.
Example: A length is recorded as 7.4 cm (to 1dp). Lower bound = 7.35 cm; Upper bound = 7.45 cm. Written formally: 7.35 ≤ L < 7.45. The upper bound is not included (it would round up to 7.5).
Upper and lower bounds appear in GCSE papers, especially in questions involving compound measures. Common exam question: "A fence panel is 3.2 m long to 1dp. What is the maximum total length of 5 fence panels?" Answer: max length per panel = 3.25 m; max total = 5 × 3.25 = 16.25 m.
For the AQA GCSE specification, you must be able to determine bounds for both addition/subtraction and multiplication/division scenarios. The key rule: for a sum, the upper bound of the sum = upper bound A + upper bound B. For a quotient, the upper bound = upper bound numerator ÷ lower bound denominator.
To round to 2 decimal places, look at the third decimal digit. If it is 5 or more, round up the second decimal digit by 1. If it is 4 or less, leave it unchanged. Example: 3.14159 → look at 1 (3rd decimal) → less than 5 → 3.14. Another example: 3.145 → look at 5 (3rd decimal) → 5 or more → 3.15. If rounding causes a carry (e.g. 9 + 1 = 10), carry the 1 to the left: 3.99 rounds to 4.00 when rounded to 1dp.
Significant figures count from the first non-zero digit. To round 0.004567 to 2 significant figures: the first significant digit is 4, the second is 5. Look at the next digit (6): round up. Result: 0.0046. For 1234 to 3 sig figs: digits are 1, 2, 3; look at 4 (next digit): round down; result: 1230. For very large numbers, use scientific notation to make the significant figures explicit.
Banker's rounding (also called round-half-to-even) rounds 0.5 exactly to the nearest even number instead of always rounding up. So 2.5 → 2, 3.5 → 4, 4.5 → 4, 5.5 → 6. This eliminates the upward bias that standard half-up rounding introduces when applied to many 0.5 values. It is the default in Python 3, IEEE 754, and many banking systems.
Rounding adjusts the last kept digit based on what follows (5-up, 4-down). Truncating removes all digits after the specified position with no adjustment whatsoever. For 3.79 to 1dp: rounded = 3.8, truncated = 3.7. Truncation always moves toward zero: −3.79 truncated to 1dp = −3.7 (not −3.8). Truncation is common in computing (integer division) but introduces systematic downward bias if used instead of rounding.
To round to the nearest 10, look at the units digit. If it is 5, 6, 7, 8, or 9, round up to the next multiple of 10. If it is 0, 1, 2, 3, or 4, round down to the current multiple of 10. Examples: 43 → 40 (units digit 3), 47 → 50 (units digit 7), 145 → 150 (units digit 5, round up), 195 → 200 (causes carry).
When a number is given to a specific accuracy, the upper and lower bounds show the range of possible original values. If L = 7.4 cm (to 1dp), then the lower bound is 7.35 cm and the upper bound is 7.45 cm, written 7.35 ≤ L < 7.45. The upper bound is excluded because 7.45 would round up to 7.5. Upper and lower bounds are a key GCSE Higher topic and appear in AQA and Edexcel exam papers.