Approximation and rounding

Starting at the left, the 1st Significant Figure of a number is the first digit which isn't 0. The subsequent Significant Figures (2nd, 3rd etc) are those that come immediately after the 1st and can be 0

You will often be asked to give an answer to 'n significant figures' where n is usually a number between 1 and 3.

If you're rounding to 2 sig.figs. say, the last digit will be the 2nd significant figure.

Sometimes you will need to round up. To decide whether to round up, we look to the n+1 significant figure. This is called the 'decider'. If the decider is 5 or more, round up If it's 4 or less, leave the digit as it is.

What is 77.029 to 3 significant figures?

The first 3 significant figures are 77.0. We now need to decide whether or not to round up. The decider is 2 so we don't. The answer is 77.0 Note, the 0 here is a significant figure so we leave it in our answer.

What is 8075.57 to 3 significant figures?

The first 3 significant figures are the 807_ We need to decide whether or not to round up. The decider is 5 so we do round up. The answer is 8080

Approximation is a technique used to get a rough idea of the result of a Calculation. It's used a lot in everyday life to estimate how much something will cost, how much of something we need to buy etc.

Round all the numbers in the calculation to ones that are easy to use and then do the calculation.

The ≈ sign denotes 'is approximately equal to' so use this when you're estimating.

And don't forget to show all your workings.

Without using a calculator, estimate 15.0201 x 5.8949

Round the figures in the calculation to whole numbers. 15.0201 x 5.8949 ≈ 15 x 6 ≈ 90.

Whenever we measure something, there is always a Limit to the Accuracy of the measurement.

The accuracy may be given in terms of decimal places, significant figures or 'the nearest x'. You just need to consider the extreme values within these limits.

If a value is given to the nearest 'x', its upper and lower bounds will be the value ± x/2

If a value is given as 98700 correct to 3 significant figures, what are its lower and upper bounds?

In this instance, the 3rd significant figure is the 100s unit so, 98700 is rounded to the nearest 100. So, the upper and lower bounds are 98700 ± 100/2 ie. 98750 and 98650

Whenever we measure something, there is always a Limit to the Accuracy of the measurement and this needs to be taken into account when calculating with rounded values.

To find the extreme values of a calculation, you just need to consider the extreme values of each term within it and what effect they have on the calculation.

For example, if you're dividing by a rounded number, the overall result will be biggest when that number is at its smallest.

A football pitch is 103m long and 46m wide. Both measurements are correct to the nearest metre. What is the maximum possible area of the pitch?

To the nearest metre means each dimension could be up to ± 0.5m out So, the maximum possible area will be 103.5 x 46.5 = 4812.75 m²

The number of Decimal Places in a number is simply the number of digits that come after the decimal point.

You will often be asked to give an answer to 'n decimal places' where n is usually a number between 1 and 3.

Sometimes you will need to round up. To decide whether to round up, we look to the digit that's n+1 places after the decimal point. This is called the 'decider'. If the decider is 5 or more, round up If it's 4 or less, leave the digit as it is.

(i) Round 4.09746 to 3 decimal places. (ii) Round 5.5555 to 2 decimal places

(i) The decider is the 4th digit after the point, ie. 4 so we don't round up. So, to 3 decimal places, 4.09746 is 4.097

(ii) The decider is the 3rd digit after the point ie. 5 so we do round up. So, to 2 decimal places, 5.5555 is 5.56

We Round numbers to make them easier to understand and use. It's something we do everyday without really thinking about it. Eg. when somebody asks how long it took to get to school you say '10 minutes' rather than '9 minutes 38 seconds'

Remember to look out for what you're rounding to. Is it the nearest metre, centimetre, kilometre?

Don't forget that 5 and above rounds up, 4 and below leaves the number the same.

What is 55,555 to the nearest thousand?

We're asked to round to the nearest thousand. We currently have 55,000 in the number. Look at the hundreds digit - that's a 5 so we round up to 56,000

To the nearest whole day, how long is 102 hours?

There are 24 hours in a day so 102 hours is 4 days and 6 hours ie. 4.25 days. To the nearest whole day, that's 4 days.