(1) Non-zero digits are always significant, as are zeroes between non-zero digits.

Example:
9,683 has four significant digits
15.60007 has 7 significant digits

 

(2) Leading zeroes before the first non-zero digit are not significant. They serve only as placeholders and do not represent measured data.

Example:
0.0005 has one significant digit

(3) Trailing zeroes right of the decimal point are always significant. They are not needed as placeholders, but represent actual measured data.

Example:
15.0000 has six significant digits
3.1560 has five significant digits

 

(4) Trailing zeroes left of the decimal point are ambiguous. They may be serving only as placeholders, or they may represent measured data. Written in this format, we can't know.

Example:
52,000 may have two, three, four, or five significant digits - we can't tell from the way it is written.

It is very poor form to report numbers with an ambiguous degree of uncertainty. In these cases, you should always use scientific notation.

Example:
5.2 x 10⁴ has two significant digits
5.2000 x 10⁴ has five significant digits

Some numbers are "exact", and can be considered to have an infinite number of significant digits. These include:

• Cardinal numbers (counting numbers): A dozen eggs contains exactly 12 eggs, not 12.000001 eggs. Eggs only come in whole numbers.

• Some mathematical relationships or constants are exact by definition. For example, the speed of light is defined as exactly 299,792,458 m/s, and there are exactly 1,000 grams in a kilogram.

When you make calculations using data with a specific level of uncertainty, it is important that you also report your answer with the appropriate level of uncertainty (i.e., the appropriate number of significant digits).

Note that significant digits are important only when reporting your final answer. You should use all available digits, both significant and insignificant, during intermediate calculations, and round to the nearest significant digit only when reporting the final result.

There are different rules for determining the appropriate number of significant digits in the results of different mathematical operations.

When quantities are multiplied or divided, the number of significant figures in the answer is equal to the number of significant figures in the quantity with the smallest number of significant figures.

Example:

1.23 * 4567.89

1.23 has three significant digits; 4567.89 has six significant digits. The result will have the smaller of these - three significant digits. Your calculator produces 5618.5047 as a result; round it to three significant digits and report 5.62 x 10³.

Caution: Do not report 5620 as your result. The last zero would be ambiguous.

When quantities are added or subtracted, the number of decimal places in the answer is equal to the number of decimal places in the quantity with the smallest number of decimal places.

Example:

1.234 + 567.89

1.234 has three digits right of the decimal point; 567.89 has two. The result will have the smaller of these - two digits right of the decimal point.

This is easier to see if you line up the figures in a column:

    1.234
+ 567.89
---------

Your calculator produces 569.124 as a result; round it to two significant digits right of the decimal point and report 569.12.

Be especially careful with numbers which are given in scientific notation.

Example:

1.2 + (3.45 x 10^-4)

The best way to solve this problem is to write the numbers in a column in ordinary notation:

  1.2
+ 0.000345
----------

Your calculator returns 1.200345, but only one digit right of the decimal is significant. Report your result as 1.2.

You may also convert all numbers into scientific notation with the same exponent:

Example:

(1.23x10⁵) + (4.56x10⁶) + (7.89x10⁷)

     1.23 x 10⁵
    45.6  x 10⁵
 + 789.   x 10⁵
----------------

The full answer would be 835.83 x 10⁵, but the last two digits are not significant. Report your result as 836. x 10⁵ or 8.36 x 10^7.

Retain in the mantissa (the number to the right of the decimal point in the logarithm) the same number of significant figures as there are in the number whose logarithm you are taking

Example:

log 9.85

Your calculator returns 0.99343623. Since the original number had three significant digits, the resulting logarithm can only have three significant digits in its mantissa (i.e., right of the decimal point), so report the result as 0.993.

Compute the number of significant digits to retain in the same order as the operations: first logarithms and exponents, then multiplication and division, and finally addition and subtraction.

It is important to keep insignificant digits during the intermediate calculations, and round to the correct number of significant digits only when reporting the final answer. In the following example, insignificant digits that are retained in intermediate calculations are shown in italics.

Example:

  (log 5.23 x 10⁵) * (1.89 + 115.7553)

First do the logarithm. Because the base number has three significant digits, the result will have three significant digits in the mantissa (right of the decimal point). For now, make a note of this, but write down the entire number.

 three sig figs in mantissa
= (5.718501689) * (1.89 + 115.7553)

Now we must do the addition before the multiplication, because of the parentheses. The sum will have two significant figures right of the decimal, but again we'll write down the whole number, since we still have the product to perform.

                 two SIG figs right of decimal
= (5.718501689) * (117.6453)

To calculate the product, we use all of the figures, but note that the first term (from the log) has four significant figures while the second term has five. Since this is a product, we keep only the lower number, or four total.

  four SIG figs
= 672.7548467529117

So report your final result as:

= 672.8   or   6.728 x 10²