Round to decimal places and significant figures

Key Rules:

• 1. To round to a number of decimal places, keep that many digits after the point and look at the very next digit: 5 or more rounds the last kept digit up, otherwise it stays.
• 2. Significant figures start at the first non-zero digit, and every digit after that counts, zeros included.
• 3. To round to significant figures, keep the stated number of significant digits, round the last one using the next digit, and fill any remaining places before the point with zeros so the number keeps its size.
• 4. When rounding pushes a 9 up, carry into the next digit: 4.596 to 2 decimal places becomes 4.60.
• 5. Keep trailing zeros that show the accuracy: 4.60 to 2 decimal places and 0.00380 to 3 significant figures are complete answers.

Remember:

• •Decimal places count from the point; significant figures count from the first non-zero digit. The next digit decides: 5 or more rounds up.

Don’t Forget...

• •Counting the leading zeros of a small decimal as significant figures, so 0.0038 is read as having four of them. Zeros before the first non-zero digit only fix the place value; significance starts at the 3, so 0.0038 has 2 significant figures.
• •Writing 20,764 to 3 significant figures as 208. The discarded digits must be replaced with zeros before the point, otherwise the number shrinks a hundredfold; the answer is 20,800.
• •Dropping the final zero and giving 4.6 as the answer for 4.596 to 2 decimal places. 4.60 states that the hundredths digit is known to be zero; 4.6 claims only 1 decimal place of accuracy.

Worked Example:

Round 20,764 to 3 significant figures.

The first significant figure is the 2, and the zero after it counts because it sits after a non-zero digit: the first three significant figures are 2, 0 and 7.

The next digit is 6, which is 5 or more, so the 7 rounds up to 8.

The digits 6 and 4 are discarded, but their places must be filled with zeros to keep the number in the ten thousands.

Answer: 20,800

Why It Works:

Rounding chooses whichever value of the stated accuracy is nearest to the original number, and the first discarded digit tells you which side of the halfway point the number sits: 5 or more means the higher neighbour is at least as close. Significant figures apply the same test but measure accuracy relative to the size of the number, which is why counting starts at the first digit that carries information.

Another Example:

Round 4.596 to 2 decimal places.

Keep two digits after the point, 5 and 9, and look at the next digit: it is 6, so round up.

Rounding the 9 up carries into the tenths: 59 hundredths becomes 60 hundredths.

Write the answer as 4.60, keeping the zero to show the accuracy is 2 decimal places.

Answer: 4.60