Rounding and Approximation

Before you can round, you have to name the places. In 3.472 the 3 is units, then straight after the point the 4 is tenths, the 7 is hundredths and the 2 is thousandths. You count decimal places starting at the point and moving right, so the first decimal place is tenths, the second is hundredths and the third is thousandths. That means rounding to two decimal places is an instruction to stop after the hundredths digit. Notice each place is ten times smaller than the one to its left, so digits further and further right matter less and less. Those tiny digits on the end are exactly the ones rounding lets you throw away.

Rounding replaces a number with the nearest tidy one. Put 6.4 on a number line between 6 and 7 and you can see it sits left of the halfway mark, so it is closer to 6 and rounds down to 6. You do not need the picture every time, because there is a rule. Look at the very next digit after the place you are keeping. If it is 0, 1, 2, 3 or 4, leave your last kept digit alone. If it is 5, 6, 7, 8 or 9, add one to that digit. Either way you then delete everything after it. Exactly 5 sits the same distance from both, so by agreement it rounds up.

Now put the rule to work. To round 3.472 to one decimal place, keep 3.4 and look at the very next digit, which is 7, so the 4 goes up to 5 and you get 3.5. To two decimal places, keep 3.47 and look at the 2, which is small, so nothing changes and the answer is 3.47. Same number, different instructions, different answers. Try 12.856: to one decimal place the next digit is 5, so 12.8 becomes 12.9, and to two places the next digit is 6, so 12.85 becomes 12.86. One warning that catches lots of people — never round twice. For 8.649 to one decimal place, work from the original: the next digit is 4, so the answer is 8.6, not 8.7.

What if the digit you have to increase is already a 9? Take 2.97 to one decimal place. You keep 2.9, the next digit is 7, so the 9 must go up — but 9 plus 1 is 10, so the 1 carries into the units and 2 becomes 3, giving 3.0. Keep that point zero, because it shows you answered to one decimal place. Sometimes the carry keeps going: 9.96 to one decimal place becomes 10.0. Whole numbers work the same way, you just cut at a different place. For 4826, the nearest ten is 4830, the nearest hundred is 4800 because the next digit is only 2, and the nearest thousand is 5000.

Rounding is not only a set of steps — you also have to choose how exact to be. Money goes to two decimal places because that is cents. A length read off a ruler marked in millimetres goes to one decimal place in centimetres, so writing 6.4173 cm is inventing precision your ruler never had. An area you calculated should match the measurements it came from, so 23.68 becomes 23.7 square centimetres. And when the answer counts real objects, round to a whole number, remembering that 17.4 buses means you actually need 18. Rounding also powers estimating: 19.6 times 4.2 is roughly 20 times 4, which is 80, so an answer of 823 is clearly wrong.