Rounding

Rounding

Rounding is just a convenient way of keeping numbers simple. Nobody wants to have to remember all the decimals in 𝝅 (which is 3.14159265358979323846 to 20 decimal places), 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, so it’s often useful to round values to the nearest power of 10, eg 10 (10¹), 100 (10²) or 1,000 (10³). 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, for instance, 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.

Sample Questions

  1. What is 5.5 to the nearest whole number?
  2. What is 46 to the nearest 10?
  3. What is 62 to the nearest 100?
  4. What is 236 to the nearest 1000?
  5. What is 500 to the nearest 1000?

Decimal Places (d.p.)

Using decimal places is helpful for long decimals that seem to go on forever. If you round to a certain number of decimal places (dp), the number must have that number of digits after the decimal point, eg 𝝅 to two decimal places (2 d.p.) is 3.14. 𝝅 is an irrational number that goes on forever, 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.

Sample Questions

  1. What is 5.24 to one decimal place?
  2. What is 18.2222… to one decimal place?
  3. What is 9.99 to one decimal place?
  4. What is 4 to 2 d.p.?
  5. What is 6.25 to three decimal places?

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, you should set to zero all the values between the last significant figure and the decimal point and ignore any decimal places, eg 1,234,567.89 to four significant figures is 1,235,000.

Values ending in lots of nines can look a bit weird when rounded, eg 9.99999 to three significant figures is 10.0…!

Sample Questions

  1. What is 13 to one significant figure?
  2. What is 7.89 to two significant figures?
  3. What is 2,495 to three significant figures?
  4. What is 1,254.36 to 2 sig. figs?
  5. What is 99.9 to 1 sig. fig.? 

     

     

     

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