Primary Mathematics/Introduction to significant digits

Significant digits are one crude way to denote a range of values. They apply to measured or estimated numbers where the quantity can vary from the provided value.

Examples

By default, significant digits are measured by

2000 means 2000 ± 500   =  1500 to 2500
200  means  200 ± 50    =   150 to 250
20   means   20 ± 5     =    15 to 25
2    means    2 ± 0.5   =   1.5 to 2.5
0.2  means  0.2 ± 0.05  =  0.15 to 0.25
0.02 means 0.02 ± 0.005 = 0.015 to 0.025

(Note: The first three examples may have more significant digits depending on context.)

A terminal decimal point can be added to change the meaning of the first three cases:

2000. means 2000 ± 0.5 = 1999.5 to 2000.5
 200. means  200 ± 0.5 =  199.5 to 200.5
  20. means   20 ± 0.5 =   19.5 to 20.5

An additional zero after the decimal point also changes the meaning:

2000.0 means 2000 ± 0.05 =  1999.95 to 2000.05
 200.0 means  200 ± 0.05 =   199.95 to 200.05
  20.0 means   20 ± 0.05 =    19.95 to 20.05
   2.0 means    2 ± 0.05 =     1.95 to 2.05

Here we've added another zero after the decimal point:

2000.00 means 2000 ± 0.005 = 1999.995 to 2000.005
 200.00 means  200 ± 0.005 =  199.995 to 200.005
  20.00 means   20 ± 0.005 =   19.995 to 20.005
   2.00 means    2 ± 0.005 =    1.995 to 2.005
   0.20 means  0.2 ± 0.005 =    0.195 to 0.205

Limitations

• Significant digits only allow for ranges where the amount that can be added to the base value and the amount that can be subtracted are the same.
• The value added and subtracted must also be 5 times or divided by a factor of 10.
• Exact values have no error or variation. Thus, $8472.35 will only mean $8472.35.

Use with scientific and engineering notation

To represent certain ranges, like 2000 ± 50, it's necessary to use either scientific or engineering notation:

0.20 × 104 Mistake: this is neither proper scientific notation nor proper engineering notation:

Proper scientific notation must have a nonzero digit to the left of the decimal point. Proper engineering notation must use multiples of 3 (e.g., 5 x 10³, 603 x 10⁶, 42.3 x 10⁹, ...)

2.0 × 103

Beyond significant digits

More complex ranges can be represented using the ± sign:

2000 ± 38.3

If the plus and minus values are different, the actual range can be listed, or separate plus and minus values can be listed. The range shown to the right of the plus and minus symbol is called an error. In this example, there are two significant digits, and one doubtful digit.

Note that the ± sign shouldn't be confused with the ± operator, which can indicate either an exact addition or subtraction. This will be covered in Algebra.

Accumulating error

As you perform mathematical operations with numbers, the error also increases.

• When you add or subtract numbers, the ranges are added together.
• When multiplying numbers, the error is converted into a percent of the main value (e.g. 20 ± 1 becomes 20 ± 5%). The percentages the two measurements added together, and the percentage is converted back to reflect the error of the original number.