Rounding

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Rounding is a process that takes a number such as 34.2198 and makes it less exact, as in 34.22. There are many reasons to do this. For example, I might get on a really accurate scale and find that I weigh 217.834 pounds. But nobody cares about the final digits of that value. Most of the time the only part of my weight measurement that someone wants to hear is my weight to the "nearest" pound. In that case I should report my weight as 218 pounds (after all, at 217.834, I am between 217 and 218 and I am "closer" to 218 than I am to 217).

On the other hand, if I am a wrestler and I am being weighed for a match then the officials will want to have a more exact reading of my weight. They my want to know my weight to the"nearest tenth" of a pound. In that case I should report my weight as 217.8 pounds (after all, at 217.834, I am between 217.8 and 217.9 and I am "closer" to 217.8 than I am to 217.9).

In order to round a number we need to know the position, the rightmost place, in the number where we want to end our "feeling of exactness". We recall that we have names for positions in numbers. Some of those names are shown in the following table where we identify the various places in the number 1437.8526.

Table 1: Place value chart for 1437.8526
place name	thousands	hundreds	tens	ones or units		tenths	hundredths	thousandths	ten thousandths
alternate name						1 decimal place	2 decimal places	3 decimal places	4 decimal places
example number	1	4	3	7	.	8	5	2	6
table column name	A	B	C	D	E	F	G	H	I

Let us say that we want to round our value, 1437.8526, to the nearest tens place. That means that we want the digit in column C in the table above to be our last "feeling of exactness" digit. Essentially, if we were counting by 10's we would "hit" 1400, 1410, 1420, 1430, 1440, 1450 and so on. Our particular value 1437.8526 would "fit" between 1430 and 1440, and it would be closer to 1440. Therefore, we say that 1437.8526 rounded to the nearest tens is 1440.

We actually have an algorithm, a sequence of steps, for doing this. First, this algorithm is stated for the case above where we are rounding to a place to the left of the decimal point. In our instance above, because we want to round to the nearest tens, our column C digit, we look at the digit in the next column to the right, the column D digit. If that column D digit is a 0, 1, 2, 3, or 4 then we

leave our column C digit as it is and
we put a 0 in column D
we change any other digits to the right, up to the decimal point, to be 0
drop any digits to the right of the decimal point.

On the other hand, if the digit in column D is a 5, 6, 7, 8, or 9 then we

increase our column C digit by 1
- if the column C digit is a 9 then increasing it by 1 would make it a 10 for which we would write the 0 in column C and "carry" the 1 back to add it to the digit in column B
we put a 0 in column D
we change any other digits to the right, up to the decimal point, to be 0
drop any digits to the right of the decimal point.

Second, this algorithm is stated for the case where we are rounding to a place to the right of the decimal point. Consider the case where we want to round 1437.8526 we want to round to the nearest hundredths place, our column G digit. We look at the digit in the next column to the right, the column H digit. If that column H digit is a 0, 1, 2, 3, or 4 then we

leave our column G digit as it is and
we drop all digits to the right of column G.

On the other hand, if the digit in column H is a 5, 6, 7, 8, or 9 then we

increase our column G digit by 1
- if the column G digit is a 9 then increasing it by 1 would make it a 10 for which we would write the 0 in column G and "carry" the 1 back to add it to the digit in column F
we drop all digits to the right of column G.

Rounding 1437.8526 to the hundreds place results in 1400.
Rounding 1437.8526 to the tens place results in 1440.
Rounding 1437.8526 to the ones place results in 1438.
Rounding 1437.8526 to the tenths place results in 1437.9.
Rounding 1437.8526 to the hundredths place results in 1437.85.
Rounding 1437.8526 to the thousandths place results in 1437.853.

Here is an example where rounding involves "carrying" a 1 to the left.
Rounding 298.3956 to the hundredths place results in 298.40.

For rounding to the "tenths", "hundredths", "thousandths", or "ten thousandths" place we often just say round to the "1 decimal", "2 decimal", "3 decima', or "4 decimal" place, respectively. For example,
Rounding 1437.8526 to 1 decimal place results in 1437.9.
Rounding 1437.8526 to 2 decimal places results in 1437.85.
Rounding 1437.8526 to 3 decimal places results in 1437.853.

Look at one more set of exmples. Consider the value 3809.9569.
Rounding 3809.9569 to the hundreds place results in 3800.
Rounding 3809.9569 to the tens place results in 3810.
Rounding 3809.9569 to the ones place results in 3810.
Rounding 3809.9569 to the tenths place results in 3810.0.
Rounding 3809.9569 to the hundredths place results in 3809.96.
Rounding 3809.9569 to the thousandths place results in 3809.957.

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©Roger M. Palay Saline, MI 48176 December, 2022