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 pi and root two. 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 zero point three recurring, pi runs on forever without repeating, and root two never settles into a neat decimal. To work with them we often round, writing pi as three point one four or root two as one point four one. 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 the whole number, here the constant pi. 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
Reusing a rounded value in a repeated calculation lets its error accumulate, growing as the number of steps grows.
Each step reuses the rounded value, so its small error is carried forward and added to again. After one step the gap is tiny; after ten it has grown roughly tenfold. The exact chain 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 one third names one exact point, sitting a third of the way from zero to one. Its rounded decimal, zero point three three three, 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 pi and root two: 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 one quarter, the decimal zero point two five, and twenty-five 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.
A fraction such as one third names an exact point on the number line. Reveal its 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 pi as three point one four one six instead of three point one four 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. Pi and root two are irrational, which means no finite decimal will ever equal them exactly. Each digit you add brings you closer, halving and halving the gap, 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 pi, right through the working.
More digits shrink the error, but never to zero
Truncating an irrational number reduces the error with each kept digit, yet no finite decimal ever equals it exactly.
Keeping more digits of pi shrinks the error fast, but it never reaches zero: pi 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 root two by root two while keeping the surd gives exactly two, with no error at all; rounding root two to one point four one first gives one point nine eight eight one, 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, root two times root two is exactly 2 with no error. Round root two 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 pi. The error in your approximation:
4. Evaluate root two times root two, keeping the surd exact.
5. When is it safest to round a real number in a multi-step calculation?