## Decimal Places and Significant Figures

Most of the time extreme accuracy is not needed, and is usually not even possible especially in our everyday life. For example, walking pace is about 4 miles per hour, and to say walking speed is 3.9876756263. is a little bit extreme. Walking speed is an average of the walking speed of many people and it is sufficient here to say 4km per hour.

Usually when approximating in this way we work out an answer using the numbers we are given, then we round the answer to so many significant figures or decimal places. Before we do this we will explain how to round numbers to so many decimal places or decimal places.

**Decimal Places**

Round 3,54681 to 4 decimal places.

The four decimal places are illustrated in green:3.54681. When we round we look at the 1 ^{st} figure past the number of places we want. If it is 4 or less, we last decimal place does not change. If it is 5 or more the last decimal place increases by one. In this case the fifth decimal number is 1 so the 4 th decimal place does not change. The answer here is 3.5468.

Round 435.35 to one decimal place.

435. 35. The second decimal place is 5 so the 3 rounds up. The answer is 435.4

Round 0.01 to one decimal place.

0. 01 The second decimal place is 1 so the 0 after the decimal point does not change. The answer is 0.0

**Significant Figures**

The first significant figure is the first non zero digit in the number starting from the left.

The first significant figures in each number below are highlighted.

234, 43.078, 0..987, 2.34*10^4, 3.45*10^-2

A zero counts as a significant figure if it occurs after the first non zero number but not if it occurs before.

The green zero here 9. 032 is a significant figure but the red zero here is not 0.210. As with decimal places, if the number just beyond the number of sigificant figures we want is 4 or less the last significant figure does not change. If the number is 5 or above, the last significant figure increases by 1.

Round 43.2 to two significant figures

43.2 The two significant figures are highlighted. The answer is 43.

Round 438 to two significant figures.

438 The two significant figures are highlighted. We round the 3 up to 4 and the 8 becomes a zero. The answer is 440. Notice that 440 is close to 438.

DON'T MAKE THE MISTAKE OF NOT WRITING DOWN THE ZEROES BEFORE THE DECIMAL POINT. THE ANSWER IS NOT 43 OR 44.

Round 43.089 to 4 significant figures.

43.089 The four significant figures are highlighted. The answer is 43.09

We may be asked: A plane travels 6454 miles in 6 and a half hours. Find the average speed to 2 significant figures (s.f.) and 4 decimal places.

Speed = distance over time =6454/6.5 = 992.9230769=990 to 2 s.f.

Some times we may be asked to round for example, 45999 to 4 s.f.

45999. The 9 just outside means the 9 just inside the number of significant figures rounds up and we have to carry. In this case the answer is 46000.