How do you round decimals?

Rounding is the one piece of arithmetic that everyone does and almost nobody does carefully. The rule takes ten seconds to learn. The mistakes come from three places: looking at the wrong digit, rounding too early, and trusting a machine that stores decimals in binary. This post covers all three.

Place value first

You cannot round to two decimal places without knowing which digit is the second decimal place. Every digit after the point has a name, and the names count from the point outward:

3 . 1 4 1 5 9
│   │ │ │ │ └ hundred-thousandths
│   │ │ │ └── ten-thousandths
│   │ │ └──── thousandths      (3rd decimal place)
│   │ └────── hundredths       (2nd decimal place)
│   └──────── tenths           (1st decimal place)
└──────────── ones

"Round to the nearest hundredth" and "round to two decimal places" mean the same thing. So does "round to the nearest cent" when the number is money.

The method

Round 3.14159 to two decimal places.

step 1  find the rounding digit
        3.14159        the 4 is in the hundredths place

step 2  look at the next digit only
        3.14159        the 1, nothing further right matters

step 3  apply the 5 rule
        1 is less than 5, so the 4 stays a 4

step 4  drop everything after the rounding place
        3.14

Same number, one more place. To three decimal places the rounding digit is the 1 in the thousandths place and the next digit is 5. Five or more rounds up, so the 1 becomes 2 and 3.14159 rounds to 3.142. The whole set:

Notice that 3.142 does not come from 3.14. It comes from 3.14159. Each row is rounded from the original, which matters more than it looks, and gets its own section below.

Only the next digit counts

The most common error is to let a long tail of digits sway the decision. It cannot. Round 7.4999 to one decimal place: the rounding digit is 4, the next digit is 9, so it rounds up to 7.5. Now round 7.4499 to one place: the rounding digit is still 4, the next digit is 4, so it stays, and the answer is 7.4. The two trailing nines are irrelevant. 7.4499 is closer to 7.4 than to 7.5, and the single next digit already knows that.

Carrying through nines

When the rounding digit is a 9 and it has to go up, it becomes 10, and the one carries left exactly as it does in addition:

2.9975  to 2 places    next digit 7, round up
        2.99 + 0.01  =  3.00

19.96   to 1 place     next digit 6, round up
        19.9 + 0.1   =  20.0

0.0996  to 3 places    next digit 6, round up
        0.099 + 0.001  =  0.100

9.999   to 2 places    next digit 9, round up
        9.99 + 0.01  =  10.00

Keep the zeros. If the question asked for two decimal places, 3.00 is the answer, not 3. The zeros say how precise the value is. A lab result of 3.00 g promises something that 3 g does not.

Money

Prices round to the nearest cent, which is two decimal places. Sales tax of 7% on a 24.99 item:

24.99 × 0.07  =  1.7493
to the nearest cent      next digit 9, round up
tax                      1.75
total   24.99 + 1.75  =  26.74

The same machinery runs the tip post and the discount post. Compute the exact amount, then round once, to cents, at the end.

What to do with a 5

The school rule says 5 rounds up: 2.5 becomes 3, 7.125 to two places becomes 7.13. It is simple and it is what most people mean by rounding. It also has a small bias, because 5 is exactly halfway, and sending every halfway case upward pushes totals up.

Banks, statisticians and most programming languages fix that with bankers' rounding, also called round half to even. An exact 5 with nothing after it rounds to whichever neighbour ends in an even digit:

        half up    half to even
2.5       3           2
3.5       4           4
4.5       5           4
0.125     0.13        0.12      (2 is even)
0.375     0.38        0.38      (8 is even)

Add the four values 1.5, 2.5, 3.5 and 4.5. The true sum is 12. Round each one half-up first and you get 2 + 3 + 4 + 5 = 14, two more than the truth. Round half to even and you get 2 + 2 + 4 + 4 = 12, which is exactly right. Over thousands of transactions that gap is real money, which is why banks care.

Two things to keep straight. First, the rules only disagree on an exact tie. 2.51 rounds to 3 under both, because it is past the halfway point, not on it. Second, neither rule is wrong. Pick one, say which, and apply it consistently.

Negative numbers

Round the size, then put the sign back. −3.7 rounds to −4 and −2.4 rounds to −2. The only awkward case is a tie: the school rule, properly stated, is round half away from zero, so −2.5 becomes −3, the same distance from zero as 2.5 becoming 3. Some software instead rounds half toward positive infinity, which turns −2.5 into −2. If a negative tie matters to you, check which one your tool does.

Round once, at the end

Rounding throws information away. Do it in the middle of a calculation and the next step works on a damaged number. Share 10 three ways:

10 ÷ 3  =  3.333...

rounded early   3.33 × 3  =  9.99
rounded late    10 ÷ 3 × 3  =  10, then round  =  10

A cent vanished. The same thing happens when a bill of 100 is split three ways at 33.33 each, which adds back to 99.99, and the bill-splitting post has the fix for that. The principle is general: carry the full value through every step and round only the number you write down.

A sneakier version is double rounding, where a number is rounded to one place and then rounded again to a shorter one:

2.4449  to 3 places  =  2.445
2.445   to 2 places  =  2.45       wrong

2.4449  to 2 places  =  2.44     next digit is 4

2.4449 is below 2.445, so it is closer to 2.44 than to 2.45. The first rounding lifted it onto the halfway mark, and the second rounding sent it the wrong way from there. This is exactly why the table of 3.14159 above rounds every row from the original and never from the row before it.

Rounding versus truncating

Truncating cuts the digits off without looking at them. It is faster and it is always wrong in the same direction:

          truncated to 2    rounded to 2
3.149       3.14              3.15
9.999       9.99              10.00
0.005       0.00              0.01

Truncation always moves toward zero, so every truncated value is too small in size. Rounding is off by at most half a unit of the last place, and the errors go both ways and tend to cancel. Some calculators truncate what they display without saying so. If you ever see 0.99999 where a 1 should be, that is usually what happened.

Decimal places versus significant figures

Decimal places count from the point. Significant figures count from the first non-zero digit. For big and small numbers they give very different answers:

0.004567   to 2 decimal places       0.00      (useless)
0.004567   to 2 significant figures  0.0046

123,456    to 2 decimal places       123,456   (nothing to do)
123,456    to 2 significant figures  120,000

Science and engineering round to significant figures because they care about relative precision. Money rounds to decimal places because a cent is a cent whether the bill is 4 or 4,000. Check which one a question is asking for before touching a digit.

Why a computer rounds 2.675 to 2.67

Round 2.675 to two decimal places. The next digit is 5, so by either tie rule the answer is 2.68: half-up because 5 goes up, half-even because 8 is even. Ask Python and it says 2.67. Ask JavaScript to round 1.005 to two places and it says 1.00. Neither is a bug in the rounding code. It is the storage. Most software keeps decimals in binary floating point, and 2.675 has no exact binary form, so what is actually stored is:

you typed    2.675
stored       2.67499999999999982236431605997495353221893310546875
rounded      2.67     the stored value is below the halfway point

you typed    1.005
stored       1.00499999999999989341858963598497211933135986328125
rounded      1.00

1.005 × 100  =  100.49999999999999    should be 100.5

The rounding happened before you asked for it, at the moment the number was typed in. This is the same fault that makes 0.1 + 0.2 come out as 0.30000000000000004. Prism stores 2.675 as exactly 2.675 and 1.005 as exactly 1.005, so 1.005 × 100 is 100.5 and 0.1 + 0.2 is exactly 0.3. It carries the full exact value through every step and only trims the digits it shows you, which is the round-once-at-the-end rule, enforced by the calculator instead of by you.

Round the real number, not the stored one

Prism works on exact decimals, so the value you round is the value you typed. Chain as many steps as you like and the full result is still there in history when you decide how many places to keep. History, percentages, fractions and scientific mode are free forever, no ads, no subscription. See the app.