Rounding is just a convenient way of keeping numbers simple. Nobody wants to have to remember all the decimals in 𝝅 (which is 3.1415926535897932384…), so people usually round it to 3.14 (or 22/7).
There are three ways of rounding numbers:
- using a power of 10
- using decimal places
- using significant figures.
Powers of 10
We use a decimal number system that’s based on powers of 10, so it’s often useful to round values to the nearest power of 10, eg 10, 100 or 1,000. In fact, when we round values to the nearest whole number, we’re still using a power of 10 as units are simply a multiple of one, which is 10⁰.
When you round to the nearest 10, say, you’re just saying which multiple of 10 is closest to your value, eg 23 is closer to 20 than 30, so 23 is 20 to the nearest 10.
But what if it’s halfway between? The number 25 is no closer to 20 than it is to 30, so what should you do? The answer is that mathematicians decided on a ‘convention’ (ie an agreed rule) that you should always round up if the next digit after the ones you need is five or more. That means 25 is 30 to the nearest 10 and not 20.
Using decimal places is helpful for long decimals that seem to go on for ever. If you round to a certain number of decimal places, the number must have that number of digits after the decimal point, eg 𝝅 to two decimal places is 3.14. 𝝅 is an irrational number that goes on for ever, so that’s quite handy!
Every time you round a value you have to check the next digit to see if it’s five or more. If it is, you have to add one to the previous digit, eg 𝝅 to three decimal places is 3.142 as the second 1 in 3.1415… is followed by a 5.
Trailing zeroes should be shown if you need them to reach the given number of decimal places, eg 10 to two decimal places is 10.00.
Significant Figures (or sig. fig.)
Using significant figures is good for both very big and very small numbers, so a lot of calculator papers in Maths will tell you to show your answers to three significant figures.
If you round to a certain number of significant figures, you start with the first digit that’s not zero and keep as many as you need, working from left to right as the bigger numbers are always on the left, eg 𝝅 to three significant figures is 3.14.
Every time you round to a certain number of significant figures, you have to check the next digit to see if it’s five or more. If it is, you have to add one to the previous digit, eg 𝝅 to five significant figures is 3.1416 as the 5 in 3.14159… is followed by a 9.
Significant figures can be confusing sometimes. If a number starts with one or more zeroes, you have to ignore them as they’re not ‘significant’ (ie important), so you’ll sometimes end up with very long strings of digits, eg 0.000025 is 0.00003 to just one significant figure even though it has six digits!
If you’re rounding a whole number to a certain number of significant figures, all the values after the last significant figure but before the decimal point should be set to zero and any decimal places ignored, eg 1,234,567.89 to three significant figures is 1,234,000.
Values ending in lots of nines can look a bit weird when rounded, eg 9.99999 to three significant figures is 10.0…!