# Which Digits are Significant?

Reference > Science > Significant FiguresBefore we can deal with rounding according to Significant Figure rules, we need to be able to recognize which digits are significant. We'll go through all of the details of recognizing significant figures in this section.

After we cover the rules, we'll give some example numbers, and work through how many sig figs each number has.

**Rule #1:** All non-zero digits are significant.

**Rule #2:** Any zero that is between two significant digits is also significant.

**Rule #3:** Leading zeros (zeroes that occur *before* the first non-zero digit) are never significant.

**Rule #4:** In a decimal value, trailing zeroes (zeroes after the last non-zero decimal place) are significant.

**Rule #5:** In a non-decimal value, trailing zeroes (zeroes after the last non-zero digit) are not significant *unless* we mark one as significant. In that case, all the trailing zeroes up to that mark are significant. How we mark a zero as significant varies from textbook to textbook, and you should consult with your teacher to determine how you should mark significant zeroes. One common method involves putting an overbar (like the repeating symbol) on the last significant zero.

Another method of implementing Rule #5 is to write your number in scientific notation, in which case, you write out the appropriate number of zeroes after the decimal, just as in Rule #4. As a matter of fact, writing numbers in scientific notation simplifies the process of determining which digits are significant; when written in scientific notation, *all* digits are significant!

For example, 3.5 x 10^{3} is equivalent to 3500, and 3.50 x 10^{3} is equivalent to 3500.

Note that we always use rules 1, and 2. Rule 3 applies to numbers that have no significant digits to the left of the decimal and 4 applies only to decimals. Rule 5 applies to non-decimal values. So which rules we use depends on whether the number has a decimal.

Now that we've covered the rules, let's work these out in some examples.

**Example One: 0.00023**

Rule 1: the 2 and 3 are significant, because they're not zeroes.

Rule 2: there are no zeroes between significant digits.

Rule 3: all of the zeroes are leading, so none of them are significant.

Rule 4: there are no trailing zeroes.

Therefore, the only significant digits are 2 and 3. This number has 2 sig figs.

**Example Two: 0.0050400**

Rule 1: the 5 and 4 are significant, because they're not zeroes.

Rule 2: there is one zero between significant digits, so it is significant as well.

Rule 3: all of the zeroes are leading, so none of them are significant.

Rule 4: there are two trailing digits; they are both significant.

Therefore, the significant digits are 5, 0, 4, 0, and 0. The number has 5 sig figs.

**Example Three: 10200**

Rule 1: the 1 and 2 are significant, because they're not zeroes.

Rule 2: there is one zero between significant digits, so it is significant.

Rule 5: no trailing zeroes are marked with an overbar, so none of them are significant.

The significant digits are 1, 0, and 2. This number has 3 sig figs.

**Example Four: 1252000**

Rule 1: the 1, 2, 5, and 2 are significant, because they're not zeroes.

Rule 2: there are no zeroes between any of the sig figs.

Rule 5: The zero in the tens place is marked as signficant. Therefore the tens place zero and the hundreds place zero are both significant.

The significant digits are 1, 2, 5, 2, 0, and 0. The number has 6 sig figs.

Alternately, this can be written in scientific notation as 1.25200 x 10^{6}, and you can readily see that this has 6 sig figs.

**Example Five: 3302000**

Rule 1: the 3, 3, and 2 are significant, because they're not zeroes.

Rule 2: the zero between the 3 and the 2 is significant.

Rule 5: None of the trailing zeroes are significant.

The significant digits are 3, 3, 0, and 2. The number has 4 sig figs.

**Example Six: 202.0030**

Note that in this number we have digits before *and *after the decimal point. This means that there are no leading zeroes, so we'll ignore rule 3.

Rule 1: the 2, 2, and 3 are significant, because they're not zeroes.

Rule 2: the zero between the twos, and the two zeroes between the two and the three are significant.

Rule 4: the trailing zero is significant.

All the digits in this number are significant. The number has 7 sig figs.

## Questions

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