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Rounding up

Rounding is a way of simplifying ugly numbers into nice numbers. For example, the number 98 is unnecessarily complex and we could simplify it to the beautiful number 100.

Similar numbers

The point of rounding is to take a number that is exact but too complex and make it a number that is similar in value and simpler. Normally we don't need to know that a car is travelling at 61.49 km/h, it is enough to know that it is travelling at 60. This process of turning 61.49 into 60 is then called rounding.

The principle of rounding

First, we'll show how to round numbers to whole tens. This means that the resulting number must be divisible by ten without remainder. The number 57 is not evenly divisible by ten, but the numbers 80 or 150 are. All numbers divisible by ten have a zero in place of the ones (the last digit).

Now the question is what number we would round 57 to. We follow the proximity procedure, of course - what are the closest numbers that are divisible by ten? We basically have two candidates, some number less than 57 and some number greater than 57. The first number that is less than 57 and divisible by ten is the number 50. On the other side, we get the number 60.

But 57 is closer to 60 than to 50, because the distance of 60 from 57 is 3 (60 − 57) and the distance of 57 from 50 is 7 (57 − 50). Therefore, 57 rounded to tens is equal to 60. The number 60 is the nearest number that is divisible by ten.

The number 57 is closer to 60 than to 50

But what happens if we are to round 55 to the nearest ten? This is because such a number is as far away from 50 as it is from 60, because it is 5 away from both. (Sometimes, mainly for technical reasons, other procedures are used.)

The number 55 is as far from 50 as it is from 60

Rounding rules

Again, we will assume rounding to whole tens.

We can round in basically two ways. The first way is to round down, i.e. to make the original number smaller. We round down when the last digit is between 0 and 4. For example, we round the number 23 to 20 because the last digit is a 3. Number 54 to 50 because the last digit is a four. The number 80 to 80. Be careful, we definitely don't round down to 70, if there is a zero, the number doesn't change.

The other way to round up is up, so the original number gets bigger. We round up when the last digit is between 5 and 9. So we round 19 to 20, 56 to 60, 88 to 90. We also round up if the number ends with a 5, so we round 15 to 20, 75 to 80, etc.

If the number ends in 0, 1, 2, 3 or 4, we round down. If it ends in 5, 6, 7, 8 or 9, we round up.

Rounding by orders of magnitude

So far, we have always rounded only to whole tens. But we can choose any other order we want to round to. By order we mean tens, hundreds, thousands...

If we round a number to hundreds, it means we want the number to be divisible by the remainder of the hundred. The number 1853 is not divisible by stems, the number 7200 is divisible by stems. Similarly for the other orders. Such numbers are always characterized by having a certain number of zeros at the end. A number divisible by ten has at least one zero at its end (it can have more, 7200 is of course also divisible by ten), a number divisible by stems has two zeros, a number divisible by thousands has three zeros, etc.

If we want to round a number to hundreds, we look for the nearest number that is divisible by one hundred. So we would round the number 389 to 400, because that number is closer to 389 than 300.

Rules for rounding by orders of magnitude

The procedure is virtually the same as rounding to tens, except that we don't look at the last digit of the number, but at the first digit, which must be zero.

For tens it is the first digit from the right, for hundreds it is the second digit from the right, for thousands it is the third digit from the right...

Let's try the procedure on the number 6482, which we'll round to the nearest hundred. The resulting number must end with at least two zeros, so we're looking at the second digit from the right. That's digit 8. Digit 8 tells us we should round up, that doesn't change. So we increase the value of the third digit from the right, that's digit 4, by one, and we zero out the last two digits - we get 6500. This is the next higher number that is divisible by 100.

Let's try rounding the same number, 6482, to the nearest thousand. We should get a number that has the last three digits zero. We're looking at the third digit from the right, 4. That tells us to round down. At this point, the value of the fourth digit from the right doesn't change, we just set the last three digits to 0 instead, so we get 6000.

Decimals

We can also round off decimal numbers, again depending on what order we are rounding to. For example, we can round the number 96.6 to whole units to 97. This works the same way for lower numbers. So let's have a number 0.327. If we want to round that number to hundredths, we look at the third number after the decimal point, we'll see that there's a seven, that's rounded up, so we round to 0.330. With decimals, we don't have to write zeros at the end. The number 3.200 is the same as 3.2. So we can rewrite the previous result after rounding to 0.33. If we wanted to round the original number to tenths, the result would be 0.3.

Rounding in sequence

If we have to round a number to an order of magnitude, we always look just one order of magnitude lower. We do not proceed by rounding to all orders one by one. For example, the number 746. If we round it to the hundreds, we get the number 700 because we are looking at the second digit from the right and that is 4 and that is rounded down.

But we could theoretically round the number to tens first, which would give us the number 750, because the digit 6 tells us to round up. Only now would we round to the hundreds - we'd get 800, not 700, because we're already rounding by digit 5, not digit 4.

This is called progressive rounding, and we don't usually use it because it's more tedious and less accurate.