What are the sources for these rounding calculations?

Ties round away from zero, the HALF_UP mode in the Java RoundingMode specification: https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/math/RoundingMode.html That page describes it as the rounding mode commonly taught at school, matching IEEE 754 roundTiesToAway.

Rounding Calculator

Round half up to a place you choose. The deciding digit is shown next to the result.

Smart Tips

• Round half up: if the deciding digit is 5 or more, round away from zero.

• The deciding digit is the first digit that is dropped. If nothing is dropped, it is 0.

• −2.5 rounded to the ones place is −3. The tie moves away from zero, not toward positive infinity.

• This is not the significant-figures page. That page uses a different rounding rule.

The Formula

if the discarded fraction ≥ 1/2, round away from zero

place The digit you keep
Thousands through millionths

deciding First discarded digit
0 when no digit is dropped

5–9 Round away from zero
2.5 to the ones place is 3

0–4 Keep the digit
2.4 to the ones place is 2

Ties round away from zero, the same rule as Java RoundingMode.HALF_UP. Negative ties do the same: −2.5 to the ones place is −3.

Key Insights

The deciding digit is visible

You can see the digit that forced the round, not only the final number.

Away from zero

A positive tie goes up. A negative tie goes more negative. Both move away from zero.

Places, not figures

You choose a place value. You do not choose a count of significant figures.

Padding

Rounding 5 to the hundredths place shows 5.00, so the place you asked for is still visible.

Frequently Asked Questions

Look at the first digit you would drop. If it is 0, 1, 2, 3, or 4, the kept digit stays. If it is 5, 6, 7, 8, or 9, round away from zero.

It is the first digit that is discarded. For 1.234 rounded to the hundredths place, the deciding digit is 4, and the result is 1.23. If the number already sits on the chosen place, the deciding digit is 0.

Ties move away from zero. −2.5 to the ones place is −3. −1.1 to the ones place is −1, because the deciding digit is 1.

No. This page rounds to a place you choose, and a tie always rounds away from zero. The significant-figures calculator counts digits and rounds a lone 5 to the even digit.

Thousands, hundreds, tens, ones, tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths, and millionths.

Ties round away from zero, the HALF_UP mode in the Java RoundingMode specification. That page describes it as the rounding mode commonly taught at school, matching IEEE 754 roundTiesToAway.

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Rounded value

Enter values to calculate

Deciding digit

Place

Note:Round half up sends a discarded fraction of one half or more away from zero. −2.5 to the ones place is −3. This is not significant-figure rounding.