## Approximations

When making calculations or taking measurements, approximations have to be made. Numbers can be approximated to a certain number of decimal places (often called dec. pl. or d.p.) or significant figures (often called sig.fig. or s.f.). Both methods involve the process of 'rounding off'.

### Significant Figures

In any number, the first non-zero digit is the first significant digit, the next digit is the second significant digit, etc.

e.g. In the number 397.2, the first significant digit is 3 and the second significant digit is 9.

When approximating a number to a given number of significant figures:

• Begin with the first non-zero digit and count in the number of significant figures required.
• Then, if the next digit is a 5 or above, add 1 on to the last significant digit.
• If it is below a 5, leave it as it is. This process is known as 'rounding off '.
 e.g. 325.6 (to 3 significant figures) 326 0.045 (to 1 significant figure) 0.05 72.9 (to 2 significant figures) 73 821 (to 1 significant figure) 800

### Decimal Places

When writing a number to a given number of decimal places:

• Begin by counting in the digits from the decimal point.
• Then, if the next digit is a 5 or above, add 1 on to the last digit.
• If the next digit is below a 5, leave it as it is.
 e.g. 7.32 (to 1 decimal place) 7.3 0.058 (to 2 decimal places) 0.06 4.6398 (to 3 decimal places) 4.640 51.69 (to 1 decimal place) 51.7

### Estimation

Before making a calculation or solving a problem, especially when using a calculator, an approximate answer should be obtained. This process helps to eliminate mistakes from incorrect keying and calculator malfunction.

Estimates can be obtained mentally by approximating the numbers in the calculation to 1 or 2 significant figures (depending on the question).

 Example Answer Calculate 3.185 x 8.693 Estimate: 3 x 9 = 27 Make an estimate first. Accurate (from calculator): 27.687205