Every measurement has some degree of uncertainty, and this limits the precision of the measurement. For example, if two melons are weighed on a bathroom scale, we might get a value of 1.4 Lb on one and 1.5 on the other, but if we use two balances (which measure with more precision) we would find that we get a measurement of 1.497 Lb on one and 1.499 on the other. With the bathroom scale, the uncertainty begins with the second digit, we therefore say the measurement has two significant figures. With the balance, the uncertainty doesn’t begin until the fourth digit, so it has four significant figures. The significant figures in a measurement are the ones that can be accurately replicated plus the first one that carries uncertainty. When we take measurements, we want to record all the significant figures, which would be all those that can be read directly plus one that can be reasonably estimated. In the example below, the measurement would be recorded as 3.42 , where the 3 and the 4 are definitely known and the 2 is estimated. This measurement would therefore have 3 significant figures, two that are known and one that is estimated. 1 2 3 4 Rules for counting significant figures: 1. All non-zero integers are significant. 2. Leading zeros are never significant. For example 0.00034 has two significant figures. 3. Captive zeros, those between non-zero digits are always significant. For example 0.000304 has three significant figures. 4. Trailing zeros are only significant if the decimal is present, otherwise they are not. For example 0.0003400 has four significant figures because the decimal is present so the trailing zeros are significant. But 3400 has only two significant figures because the decimal is absent so the trailing zeros not significant. When multiplying and dividing with numbers obtained from measurements, you must always round your answer to the same number of significant figures as the measurement with the least number or significant figures. Example: 173 ÷ 14.01 = 12.34832 which rounds to 12.3 The answer must be rounded to three significant figures since 173 has only three significant figures. Rounding Rules: 1. Carry all the digits through all calculations and round for significant figures only at the end. 2. In order to round a digit, look at only the one immediately to the right of it. If that number is less than 5 the rounded digit stays the same. Example: if we round 2.5649 to three significant figures we will either keep the 6 or round it up (we never round down), the number to the left of 6 is 4 so we keep the 6 and round to 2.56 If we round 2.5681 to three significant figures the 6 will be rounded to 7 and we get 2.57 It is important to note that exact numbers are assumed to have an infinite number of significant figures and have no influence on rounding. Exact numbers include numbers obtained by counting rather than by measuring (example: 24 apples), or defined numbers (1 inch = 2.54 cm).