Approximation versus Exact: How Rounding Errors Grow
ACARA v9 CONTENT DESCRIPTION “recognise the effect of using approximations of real numbers in repeated calculations and compare the results when using exact representations”
Builds on: Real Numbers: Rational and Irrational (AC9M9N01). This unit builds on knowing that real numbers include both neat fractions and never-ending irrationals such as π and √2. Here we look at what happens when those numbers are rounded for convenience and then used again and again, which is exactly the situation in measurement, finance, and any calculation that runs in stages.
Exact numbers and their rounded stand-ins
Many of the numbers that matter most cannot be written out in full. One third is 0.333..., with the threes recurring forever, π runs on forever without repeating, and √2 never settles into a neat decimal. To work with them we often round, writing π as 3.14 or √2 as 1.41. That rounded form is an approximation: a convenient stand-in that is close to the true value but not equal to it. The moment we round, we accept a small error. For a single, final answer that error is usually harmless. The interesting question, and the one this unit is really about, is what happens to that error when the rounded value is not the end of the story but the input to the next step.
Round once, or keep it exact
Rounding a real number to a few decimal places already introduces a small error, before any further working.
An exact representation keeps every digit of the number, here the constant π. Round it to two places to see the gap the rounding opens up.
Why repeated calculations let the error grow
Think of a recipe that you scale up, a balance that earns interest month after month, or a measurement that you feed through several formulas in a row. Each stage takes the result of the last one as its starting point. If you rounded early and have been reusing that rounded value, then every stage inherits the small error and adds a little more of its own. The error does not stay still; it is carried forward and compounded, growing roughly in step with the number of stages. A gap of a thousandth after one step can become a hundredth after ten. The exact calculation, by contrast, carries no error to pass on, so its stages stay true no matter how many there are. This is the heart of the matter: rounding once is a small decision, but rounding once and then reusing the result many times is a decision that grows.
The error compounds step by step
A wheel 1 metre across moves π metres each turn. Add up the turns with π kept exact, and again with 3.14 added each turn, and watch the rounded total fall behind.
Each turn adds 3.14 metres instead of π metres, so the rounded total falls about 0.0016 metres further behind with every turn. After ten turns the gap has grown tenfold, to about 0.016 metres, or 1.6 centimetres. The exact chain, 10π metres, has no gap at all.
Exact forms name a point that decimals only approach
It helps to picture these numbers on a number line. A fraction such as 1/3 names one exact point, sitting a third of the way from 0 to 1. Its rounded decimal, 0.333, lands close to that point but just short of it, because the true value keeps going with more threes forever. The same is true of π and √2: the exact symbol marks the true spot, while any rounded decimal stops a hair away. Different exact forms can even name the same point, so 1/4, the decimal 0.25, and 25 per cent are three spellings of one place on the line. Recognising that an exact form and its rounded decimal are not the same number is what makes the rest of this idea click.
Exact forms hit the point; rounded ones miss
A fraction or surd names an exact point, while its rounded decimal lands just short of that point. The gap is too small to see on the full line, so look in the zoomed window.
A fraction such as one third names an exact point on the number line. On the full line any rounded decimal looks as if it sits on top of it, so the window below zooms in 500 times. Reveal the rounded decimal to see that 0.333 stops just short of the true position.
More digits help, but never finish the job
A fair response is to keep more digits. Writing π as 3.1416 instead of 3.14 shrinks the error sharply, and for most practical work a handful of digits is plenty. But there is a limit that no amount of patience can pass. π and √2 are irrational, which means no finite decimal will ever equal them exactly. Each digit you add brings you closer, cutting the gap about tenfold on average, yet the gap never closes to nothing. So keeping more digits is a sensible way to control error, but it is not the same as being exact. If a result must be perfectly correct, the only sure route is to keep the exact form, the fraction or the surd or the symbol π, right through the working.
More digits shrink the error, but never to zero
Rounding an irrational number to more decimal places shrinks the error with each extra digit, yet no finite decimal ever equals it exactly.
Keeping more digits of π shrinks the error fast, but it never reaches zero: π is irrational, so no finite decimal equals it exactly. Rounding is always an approximation, however many digits you keep.
Keeping it exact, and rounding only at the end
The practical rule that falls out of all this is simple. Work with exact forms for as long as you can, and round only at the very end, once, to the precision the situation needs. Multiplying √2 by √2 while keeping the surd gives exactly 2, with no error at all; rounding √2 to 1.41 first gives 1.41 × 1.41 = 1.9881, already adrift from the true answer. Keeping the exact form costs nothing and protects the result. When a decimal answer is finally required, round the exact result you have reached, rather than a value you rounded several steps ago. That single habit, exact through the middle and rounded only at the end, is what keeps a long calculation honest.
Keep the surd, or round too soon
Working with an exact surd keeps the result exact, while rounding it before multiplying introduces an avoidable error.
Kept exact, √2 × √2 is exactly 2 with no error. Round √2 to 1.41 first to see how the answer drifts away from 2.
Quick self-check
1. A calculation reuses a value ten times. Rounding that value early, instead of keeping it exact, will usually make the final error:
2. Which of these is an exact representation of the number, with no rounding error?
3. You keep more and more decimal digits of π. The error in your approximation:
4. Evaluate √2 × √2, keeping the surd exact.
5. When is it safest to round a real number in a multi-step calculation?