# Significant Figures: 6.0 Rules

Significant figures tell you how accurately a number is known. This invariably depends on the precision of your instruments.

To illustrate this, use a pencil and a ruler to draw a square with sides of 8.109435 cm in length. Now, calculate the area of the square that you’ve drawn.

A ruler can only measure length to within ±0.1 cm. Our square therefore has sides 8.1 cm in length (not 8.109435 cm) because our measurements are limited by the accuracy of the ruler. The area of our square is therefore 8.1×8.1=66 cm², not 65.762936019225 cm², because there was no way to measure all of those decimal places precisely using a ruler.

Accurately-known digits are known as significant digits. All other digits are described as not significant. We must always round our final answer (not the intermediate steps) to the correct number of significant digits by following the six rules below.

### 1. Numbers without a decimal point

1. First non-zero digit is significant
2. Last non-zero digit is significant
3. All digits in-between are significant
• 45 is to 2 significant figures (s.f.)
• 1,240 (3 s.f.)
• 68,686,000 (5 s.f.)

### 2. Numbers with a decimal point

1. First non-zero digit is significant
2. All digits afterwards are significant
• 1.2 (2 s.f.)
• 6.810 (4 s.f.)
• 900,001 (6 s.f.)

### 3. Scientific notation

Scientific notation is a way of writing numbers in the form:

a × 10 where 1 ≤ a < 10.

Count the number of significant figures in a to find the number of significant figures in the number (a × 10b).

• 5.56 × 103 is to 3 significant figures
• 2.012 × 10-4 is to 4 significant figures

### 4. Conversions

Some unit conversions are exact and are said to have an unlimited number of significant figures.

• 1 minute = 60.0000000000… seconds (infinite s.f.)
• 1 metre = 100.0000000000… metres (infinite s.f.)

Temperatures usually have 3 (sometimes 4) significant figures when converted into Kelvin!

• 10°C = 283 K (3 significant figures)
• 100°C = 373 K (3 significant figures)
• 4000°C = 4273 K (4 significant figures)

Rule: Always round your final answer (not any intermediate answers!) to the smallest number of decimal places.

• 441 + 65.42 = 506 (use zero decimal places)
• 200.1 – 144.2456 = 55.9 (use 1 decimal place)

### 6. Multiplication and division

Rule: Always round your answer so it has the same number of significant figures as the input value with the smallest number of significant figures.

• 481.56 × 14.5 = 6980 (use only 3 s.f.)
• 7800 ÷ 41.1 = 190 (use only 2 s.f.)

Remember to round your ANSWER (not the intermediate steps) to the correct number of significant figures.