*e*, √2), or ratios (such as 1/2, 3/4, or 1/3), which are considered exact, and effectively have an infinite number of significant figures.

Now let's look at some examples of (presumably) measured quantities that

*do*need to have their significant figures counted:

0.0179 in (25-gauge wire diameter),

0.4600 in (4/0-gauge wire diameter),

4,186.01 Hz (highest piano key frequency),

27.500 Hz (lowest piano key frequency),

4,350 m (Mount Evans elevation),

200 m (Hamwolsan Mountain elevation).

Given that the measurements above are neither exact mathematical quantities, defined quantities, nor discrete/countable quantities, we

*can*discuss the number of significant figures these values have.

The first two measurements (0.0179 in and 0.4600 in) are both quantities that

*are*less than 1, so the flowchart rule is to count

*all*digits from

*left-to-right*, starting from the first non-zero digit on the

*left*.

So for 0.0179 in (where the digits in green denote significant figures), we start counting

*all*digits starting from the "1" (the first non-zero digit on the left), then continue counting from left-to-right the "7" and "9," and thus this measurement has

*three*significant digits. If we were to rewrite this in scientific notation, then it would be expressed as 1.79×10

^{-2}in. Note that scientific notation always shows the correct number of significant digits.

For 0.4600 in, we start counting

*all*digits starting from the "4" (the first non-zero digit on the left), and then continue counting from left-to-right the "6," "0," and "0," and thus this measurement has

*four*significant digits. If we were to rewrite this in scientific notation, then it would be expressed as 4.600×10

^{-1}in, which again shows the correct number of significant digits.

Now the next two measurements of 4,186.01 Hz and 27.500 Hz are both greater than 1, and both include the decimal point. The flowchart rule is to simply count

*all*digits. Thus 4,186.01 Hz has

*six*significant figures, and 27.500 Hz has

*five*significant figures, and the respectively would be expressed in scientific notation as 4.18601×10

^{3}Hz and 2.7500×10

^{1}Hz, each of which display their correct number of significant digits.

The next-to-last measurement of 4,350 m is greater than 1, has no decimal point, but has more than one non-zero digit, so the flowchart rule is to count

*all*digits from

*right-to-left*, starting from the first non-zero digit on the

*right*. Thus for 4,350 m, we start counting

*all*digits starting from the "5" (the first non-zero digit on the right), then continue counting the "3" and "4," and thus this measurement has

*three*significant digits. If we were to rewrite this in scientific notation, then it would be expressed as 4.35×10

^{3}m, as it only has three significant figures.

The last measurement of 200 m is greater than 1, has no decimal point, and does

*not*have more than one non-zero digit (in fact, it only has one non-zero digit). The flowchart rule is then to apply the

*"rule of two"*--where we would arbitrarily assume that such a measurement would have two significant digits, and thus express this in scientific notation as 2.0×10

^{2}m.

*significant figures*as the quantity with the fewest number of

*significant figures*.

Let's consider two measurements, 464.0 in and 1.40 in. Verify for yourself that they have four significant figures and three significant figures, respectively. If we were to multiply these two numbers together (to find an area expressed in units of square inches, or in

^{2}), then:

464.0 in × 1.40 in = 649.60 in

^{2},

but this result should only have three significant figures, as this was determined by the 1.40 in quantity, which has the fewest number of significant figures (in this case, three). In order to unambiguously express this result with the proper number of significant figures, we will round this result in scientific notation to 6.50×10

^{2}in

^{2}. (Note that expressing this result instead as "650 in

^{2}" would be ambiguous, as from the flowchart this would seem to have only

*two*significant figures, from applying the "rule of two.")

Now if we were to instead divide the two measurements 464.0 in and 1.40 in (to obtain a unitless ratio), then:

(464.0 in)/(1.40 in) = 331.4285714,

but this result should again only have three significant figures, as determined by the 1.40 in quantity, which has the fewest number of significant figures (in this case, three). In order to unambiguously express this result with the proper number of significant figures, we will truncate this result in scientific notation to 3.31×10

^{2}.

Let's add together the two measurements 464.0 in and 1.40 in. We'll write them as a "stack," aligning them properly for addition:

464.0 | in | |

+ | 001.40 | in |

= | ? | in |

Since only the hundreds, tens, ones, and one-tenths columns all have non-blank digits in the "stack," they will all be significant in the final answer. However, the last "0" in 1.40 in, in the one-hundredths column is by itself in the "stack," and so the result in that column (which would mathematically be "0") is

*not*significant.

464.0 | in | |

+ | 001.40 | in |

= | 465.40 | in |

Thus the result has only one decimal place, which will then overall have three significant figures, expressed as 465.4, and

*not*465.40.

The subtraction operation would be similar, again first writing them as a "stack":

464.0 | in | |

– | 001.40 | in |

= | ? | in |

Since only the hundreds, tens, ones, and one-tenths columns all have non-blank digits in the "stack," they will all be significant in the final answer. However, the last "0" in 1.40 in, in the one-hundredths column is by itself in the "stack," and so the result in that column (which would mathematically be "0") is

*not*significant.

464.0 | in | |

– | 001.40 | in |

= | 462.60 | in |

Thus the result has only one decimal place, which will then overall have three significant figures, expressed as 462.6, and

*not*462.60.

^{1}ft

^{2}").

Thus before performing the multiplication/division rule or the addition/subtraction rule, it will be important to know whether the quantities involved are exact counts of discrete quantities, or are actual measurements with "significant" significant figures and/or decimal places.

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