Rounding to significant figures and decimal places

A grain of rice has a mass of nought point nought nought six eight four seven two grams. Two different instructions could be put to that number. Round it to two decimal places, or round it to two significant figures. One instruction gives nought point nought one grams. The other gives nought point nought nought six eight grams. By the end of this video you will know which gives which, and why. The two rules sound almost identical, but they start counting from different places. Decimal places are the place to begin, because they count from a fixed point.

Decimal places count the digits after the decimal point, and nothing else. Take three point eight four seven, rounded to one decimal place. One digit after the point is kept, and that digit is the eight. Next, look at the digit straight after the last one you keep. Here that is the four. Five or more rounds the kept digit up. Four or less leaves it as it is. The reason is the halfway mark. Three point eight four seven sits between three point eight and three point nine, and halfway is three point eight five. A four in that spot means the number has not reached halfway. That gives three point eight, to one decimal place. Now round twenty-four point one five seven to one decimal place. Twenty-four point two. You keep the one after the point, and the digit after it is a five. Five rounds up, which turns the one into a two.

Significant figures count from a different place. A significant figure is a digit that tells you the size of the number, and the count starts at the first digit that is not zero. Once the count has started, a zero between two other digits counts too. Zeros in front of the first non-zero digit never count. They only hold the other digits in their places. Back to three point eight four seven, this time to two significant figures. The first non-zero digit is the three, which makes it the first significant figure. The eight is the second. The digit after the eight is the four again, which leaves the eight alone. Two significant figures gives three point eight, the same answer as one decimal place. They agree here only because the first significant figure sits straight before the point. Move the point, and the two rules split apart. Now put twenty-four point one five seven through the significant figures rule instead. To two significant figures, is it twenty-four point two, twenty-four, or twenty? Twenty-four. The first significant figure is the two, the second is the four, and the digit after it is a one. One is less than five, which leaves twenty-four. Same number, two instructions, two different answers. Twenty-four point two to one decimal place, and twenty-four to two significant figures. Significant figures work on big numbers too. Take forty-eight thousand three hundred and sixty, rounded to one significant figure. The first significant figure is the four, and the next digit is an eight. Eight rounds up, so the four becomes a five. Every digit after it becomes a zero, because those zeros hold the five in the ten-thousands place. That gives fifty thousand. One examiner's report on a Foundation paper puts the mix-up in one sentence. Others misunderstood rounding to one significant figure as rounding to one decimal place and incorrectly worked with eight point three four as eight point three and ten point two one as ten point two. To one significant figure, those two numbers are eight and ten. The fix is a habit: read the last words of the instruction, places or figures, before you touch the number.

Now the grain of rice, nought point nought nought six eight four seven two grams, to two significant figures. It starts with zeros, and those zeros do not count. They only place the six in the thousandths column. The first non-zero digit is the six, so the count starts there. The six is the first significant figure, and the eight is the second. The digit after the eight is a four, which leaves the eight alone. That gives nought point nought nought six eight grams, to two significant figures. The zeros in front stay in the answer. Without them it would read nought point six eight, and the mass would be a hundred times too big. The other instruction from the start asks for two decimal places. Two decimal places keeps two digits after the point, and both are zeros. The digit after them is a six, which rounds the second zero up to a one. That gives nought point nought one grams. Now round the same mass to three decimal places. Seven thousandths of a gram. Three decimal places stops at the six, and the eight behind it lifts the six up to a seven. A different examiner's report, also on a Foundation paper, records another slip with significant figures. A common incorrect response was twenty-five, showing they thought two significant figures meant the first two digits. Those students copied two digits and stopped. The habit that fixes it is to start at the first digit that is not zero, round with the digit after, and keep any zeros that hold the size.

One more rule finishes the job. Once your answer has the accuracy the question asked for, you stop rounding. A plank is ninety centimetres long, correct to one significant figure. A student is asked for its length to one significant figure, writes one hundred centimetres, and that answer is marked wrong. Why is one hundred centimetres wrong, to one significant figure? Nothing needed doing. Ninety has one significant figure already, the nine, and its zero only holds the nine in the tens place. Rounding again pushed a finished value up to one hundred. The check takes one look. Hold your answer against the instruction, and if it already has the places or figures asked for, it is finished.

To close, one rule to say back, then two numbers this video never rounded. Where does the count start for significant figures? At the first non-zero digit. Any zeros in front of it only hold its place. Next, round nought point nought four nine two to one significant figure. Nought point nought five. The four is the only significant figure kept, and the nine behind it pushes the four up to a five. Here's another: seven thousand eight hundred and fifty, to two significant figures. Seven thousand nine hundred. Two figures means the seven and the eight, the fifty lifts the eight up to a nine, and zeros fill the rest. And the grain of rice from the start comes out as nought point nought one grams to two decimal places, and nought point nought nought six eight grams to two significant figures.

Once you can read any rounding instruction and pick the right rule, give this video a thumbs-up. Scrolling back through the chapter later, that thumb tells you that you own this one. If it has not settled yet, save it to a playlist and leave the thumb off, so it waits on your list. Nobody gets this first time.

Next in the chapter is Estimating a calculation by rounding, where every number in a sum is rounded before you calculate.

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