pi: in the examples below.

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Rounding Numbers

When you have to round a number, you are usually told

how to round it. It's simplest when you're told how many

"places" to round to, but you should also know how to

round to a named "place", such as "to the nearest

thousand" or "to the ten-thousandths place". You may

also need to know how to round to a certain number of

significant digits; we'll get to that later.

In general, you round to a given place by looking at the digit one place to the right of the "target" place. If the digit is a five or greater, you round the target digit up by one. Otherwise, you leave the target as it is. Then you replace any digits to the right with zeroes (if they are to the left of the decimal point) or else you delete the digits (if they are past the decimal point).

I'll use the first few digits of the decimal expansion of

pi

:

3.14159265...

in the examples below.

• Round

pi

to five places.

"To five places" means "to five decimal places". First, I count out the five decimal places, and then I look at the sixth place:

3.14159 | 265...

I've drawn a little line separating the fifth place from the sixth place. This can be a handy way of "keeping your place", especially if you are dealing with lots of digits.

The fifth place has a

9

in it. Looking at the sixth place, I see that it has a

2

in it. Since

2

is less than five, I won't round the

9

up; that is, I'll leave the

9

as it is. In addition, I will delete the digits after the

9

. Then

pi

, rounded to five places, is:

3.14159

• Round

pi

to four places.

First, I go back to the original number (not the one I just rounded in the previous example). I count off four places, and look at the number in the fifth place:

3.1415 | 9265...

The number in the fifth place is a

9

, which is greater than

5

, so I'll round up in the fourth place, truncating the expansion at four decimal places. That is, the

5

becomes a

6

, the

9265...

part disappears, and

pi

, rounded to four decimal places, is:

3.1416

• Round

pi

to three places.

First, I go back to the original number (not the one I just rounded in the previous example). I count off three decimal places, and look at the digit in the

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fourth place:

3.141 | 59265...

The number in the fourth place is a

5

, which is the cut-off for rounding: if the number in the next place (after the one you're rounding to) is

5

or greater, you round up. In this case, the

1

becomes a

2

, the

59265...

part disappears, and

pi

, rounded to three decimal places, is:

3.142

This rounding works the same way when they tell you to round to a certain named place, such as "the hundredths place". The only difference is that you have to be a bit more careful in counting off the places you need. Just remember that the decimal places count off to the right in the same order as the counting numbers count off to the left. That is, for regular numbers, you have the place values:

...(ten-thousands) (thousands) (hundreds) (tens) (ones)

For decimal places, you don't have a "oneths", but you do have the other fractions: (decimal point) (tenths) (hundredths) (thousandths) (ten-thousandths)... For instance:Copyright © Elizabeth Stapel 2000-2013 All Rights Reserved

• Round

pi

to the nearest thousandth.

"The nearest thousandth" means that I need to count off three decimal places (tenths, hundredths, thousandths), and then round:

3.141 | 59265...

Then

pi

, rounded to the nearest thousandth, is

3.142.

• Round

2.796

to the hundredths place.

The hundredths place is two decimal places, so I'll count off two decimal places, and round according to the third decimal place:

2.79 | 6

Since the third decimal place contains a

6

, which is greater than

5

, I have to round up. But rounding up a

9

gives a

10

. In this case, I round the

79

up to an

80

:

2.80

You might be tempted to write this as "

2.8

", but, since you rounded to the hundredths place (to two decimal places), you should write both decimal places. Otherwise, it looks like you rounded to one decimal place, or to the tenths place, and your answer could be counted off as being incorrect.

Rounding and Significant Digits

Another consideration in rounding is when you have to

round to "an appropriate number of significant digits".

What are significant digits? Well, they're sort of the

"interesting" or "important" digits. For example,

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1000

has one significant digit (only the

1

is interesting; you don't know anything for sure about the hundreds, tens, or units places; the zeroes may just be placeholders; they may have rounded something off to get this value)

1000.0

has five significant digits (the "

.

0

" tells us something interesting about the presumed accuracy of the measurement being made: that the measurement is accurate to the tenths place, but that there happen to be zero tenths)

0.00035

has two significant digits (only the

3

and

5

tell us something; the other zeroes are placeholders, only providing information about relative size)

0.000350

has three significant digits (that last zero tells us that the measurement was made accurate to that last digit, which just happened to have a value of zero)

1006

has four significant digits (the

1

and

6

are interesting, and we have to count the zeroes, because they're between the two interesting numbers)

560

has two significant digits (the last zero is just a placeholder)

560 .

(notice the "point" after the zero) has three significant digits (the decimal point tells us that the measurement was made to the nearest unit, so the zero is not just a placeholder)

560.0

has four significant digits (the zero in the tenths place means that the measurement was

made accurate to the tenths place, and that there just happen to be zero tenths; the

5

and

6

give useful information, and the other zero is between significant digits, and must therefore also be counted) If you need to express your answer as being "accurate to" a certain place, here's how the language works with the above examples:Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

3.14159

is accurate to the hundred-thousandths place

1000

is accurate to the thousands place

1000.0

is accurate to the tenths place

0.00035

0.000350

is accurate to the millionths place (note the extra zero)

1006

is accurate to the units place

560

is accurate to the tens place

560 .

is accurate to the units place (note the decimal point)

560.0

is accurate to the tenths place Here are the basic rules for significant digits: 1) All nonzero digits are significant.

2) All zeroes between significant digits are significant.

3) All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are themselves significant.

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• Round

742,396

to four, three, and two significant digits:

742,400

(four significant digits)

742,000

(three significant digits)

740,000

(two significant digits)

• Round

0.07284

0.0728

0.073

• Round

231.45

231.5

231

230

Significant Digits: Additional Details

The real question comes in how to round answers to the "appropriate" number of significant digits.

The idea is this: Suppose you measure a block of wood. The length is

5.6

inches, the width is

4.4

inches, and the thickness is

1.7

inches, at least as best you can tell from your tape measure. To find the volume, you would multiply these three dimensions, to get

41.888

cubic inches. But can you really, with a straight face, claim to have measured the volume of that block of wood to the nearest thousandth of a cubic inch?!? Not hardly! Each of your measurements was accurate (as far as you can tell) to two significant digits: your tape was marked off in tenths of inches, and you wrote down the closest tenth of an inch that you could see. So you cannot claim five decimal places of accuracy, because none of your measurements exceeded two digits of accuracy. You can only claim two significant digits in your answer. In other words, the "appropriate" number of significant digits is two, and you would report (in your physics lab report, for instance) that the volume of the block is

42

cubic inches, approximately

Rounding Addition

How do you round when they give you a bunch of numbers to add? You would add (or subtract) the numbers as usual, but then you would round the answer to the same decimal place as the least-accurate number.

• Round to the appropriate number of significant digits:

13.214 + 234.6 + 7.0350 + 6.38

Looking at the numbers, I see that the second number,

234.6

, is only accurate to the tenths place; all the other numbers are accurate to a greater number of decimal places. So my answer will have to be rounded to the tenths place:

13.214 + 234.6 + 7.0350 + 6.38 = 261.2290

13.214 + 234.6 + 7.0350 + 6.38 =

261.2

Here's another example:

• Round

1247 + 134.5 + 450 + 78

to the appropriate number of significant digits.

•

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because

450

is only accurate to the tens place. First, I add in the usual way:

1247 + 134.5 + 450 + 78 = 1909.5

...and then I round my result to the tens place:

1247 + 134.5 + 450 + 78 =

1910

Rounding Multiplication

How do you round, when they give you numbers to multiply (or divide)? You would multiply (or divide) the numbers as usual, but then you would round the answer to the same number of significant digits as the least-accurate number.

• Simplify, and round to the appropriate number of significant digits:

16.235 × 0.217 × 5

First, I note that

5

has only one significant digit, so I will have to round my final answer to one significant digit. The product is:

16.235 × 0.217 × 5 = 17.614975

...but since I can only claim one accurate significant digit, I will need to round

17.614975

to

20

, which is accurate to one significant digit.

16.235 × 0.217 × 5 =

20

• Find the product of

0.00435

and

4.6

to the appropriate number of digits.

First I multiply:

0.00435 × 4.6 = 0.02001

Looking at the original numbers, I see that

4.6

has only two significant digits, so I will have to round

0.02001

to two significant digits. The

2

is the first significant digit, so the

0

following it will have to be the second significant digits. In other words, I must report the answer as being:

0.00435 × 4.6 =

0.020

The answer should not be

0.02

, because

0.02

has only one significant digit; namely, the "

2

". The trailing zero in

0.020

indicates that "this is accurate to the thousandths place, or two significant digits", and is therefore a necessary part of the answer.

Just remember the difference:

For adding, use "least accurate place".

For multiplying, use "least number of significant digits".

Another consideration in rounding is when you have to round to "an appropriate number of significant digits". What are significant digits? Well, they're sort of the "interesting" or "important" digits. For example,

3.14159

has six significant digits (all the numbers give you useful information)

1000

has one significant digit (only the

1

is interesting; you don't know anything for sure about the hundreds, tens, or units places; the zeroes may just be placeholders; they may have rounded something off to get this value)

1000.0

has five significant digits (the "

.

0

" tells us something interesting about the presumed accuracy of the measurement being made: that the measurement is accurate to the tenths place, but that there happen to be zero tenths)

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0.00035

has two significant digits (only the

3

and

5

tell us something; the other zeroes are placeholders, only providing information about relative size)

0.000350

has three significant digits (that last zero tells us that the measurement was made accurate to that last digit, which just happened to have a value of zero)

1006

has four significant digits (the

1

and

6

are interesting, and we have to count the zeroes, because they're between the two interesting numbers)

560

has two significant digits (the last zero is just a placeholder)

560 .

(notice the "point" after the zero) has three significant digits (the decimal point tells us that the measurement was made to the nearest unit, so the zero is not just a placeholder)

560.0

has four significant digits (the zero in the tenths place means that the measurement was

made accurate to the tenths place, and that there just happen to be zero tenths; the

5

and

6

give useful information, and the other zero is between significant digits, and must therefore also be counted) If you need to express your answer as being "accurate to" a certain place, here's how the language works with the above examples:Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

3.14159

1000

is accurate to the thousands place

1000.0

is accurate to the tenths place

0.00035

0.000350

is accurate to the millionths place (note the extra zero)

1006

is accurate to the units place

560

is accurate to the tens place

560 .

is accurate to the units place (note the decimal point)

560.0

is accurate to the tenths place Here are the basic rules for significant digits: 1) All nonzero digits are significant.

2) All zeroes between significant digits are significant.

3) All zeroes which are both to the right of the decimal point and to the right of all non-zero significant digits are themselves significant.

Here are some rounding examples; each number is rounded to four, three, and two significant digits.