SIGNIFICANT FIGURES and Related Concepts
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Some numbers are exact.
I have [beautiful math coming... please be patient] $\,\bf{1}\,$ daughter and $\,\bf{2}\,$ ears.
There are [beautiful math coming... please be patient] $\,\bf{100}\,$ centimeters in a meter (by definition).
The number  π  is the ratio of the circumference to the diameter of any circle.
The number [beautiful math coming... please be patient] $\,\sqrt{2}\,$ is the unique nonnegative number which, when multiplied by itself, gives $\,2\,$.

Most of the numbers used in mathematics are assumed to be exact.
For example, when a word problem says that ‘Carol bought one-half pound of chocolate’
you don't usually worry about whether it was really [beautiful math coming... please be patient] $\,0.49\,$ pounds or $\,0.52\,$ pounds.

In the sciences, however (like chemistry, physics, and biology), numbers are often suspect.
Whenever a measurement is made, the number that you get as a result has potential error associated with it.
The amount of error depends on things like the quality of the measuring instrument and the skill of the person making the measurement.

The reliability of a measurement has two components:  precision and accuracy.

Precision refers to how closely measurements of the same quantity agree with each other.
Accuracy refers to how closely measured values agree with the correct (true) value.

Measurements can be both precise and accurate.
This happens when the measurements are close to each other (precise), and also close to the true value (accurate).

Measurements can be neither precise nor accurate.
This happens when the measurements are not close to each other (not precise), and also not close to the true value (not accurate).

Measurements can be precise, but not accurate.
This happens when the measurements are close to each other (precise), but not close to the true value (not accurate).

Measurements can theoretically be accurate, but not precise.
This happens when the measurements are close to the true value (accurate), but not close to each other (not precise).
(This is hard to visualize, because if points are close to a true value, then they're usually also close to each other!)

Given a set of measurements, the average value is taken as the best value.
The range of a set of measurements is the difference between its greatest and least values.
Range is a measure of precision, since it is a measure of how close the individual measurements are to each other.

For example, suppose the following five length measurements are made (all units of feet): [beautiful math coming... please be patient] $\,7.1\,$, $\,6.8\,$, $\,6.7\,$, $\,7.3\,$, and $\,6.6\,$.

The average value is [beautiful math coming... please be patient] $\,\displaystyle\frac{7.1+6.8+6.7+7.3+6.6}{5} = 6.9\,$.

The range is the difference between the greatest value ($\,7.3\,$) and the least value ($\,6.6\,$), so the range is [beautiful math coming... please be patient] $\,7.3 - 6.6 = 0.7\,$.

Scientists communicate the precision of measurements using a concept called significant figures or significant digits.
Roughly, the number of significant figures is the number of digits believed to be correct by the person doing the measuring;
usually, there is one estimated digit.

The number of significant figures is determined using the following rules:

When significant figures are used in calculations, it is important that the result reflects the appropriate skepticism of the component numbers!
In general, computations are done as if the numbers are exact, and then the answers are rounded according to the following rules:

Notice that for addition/subtraction, the number of decimal places is the key concept,
whereas for multiplication/division, the number of significant figures is the key concept.

For computations with significant figures, you always round as the last step.
If you need some practice with rounding, click here.

Some people adopt the convention that if the number being rounded
is exactly halfway between the two rounding candidates, then you round UP.
With this convention, [beautiful math coming... please be patient] $\,2.35\,$ rounds to $\,2.4\,$, and $\,2.65\,$ rounds to $\,2.7\,$ (when rounding to one decimal place).
If this type of situation occurs a lot, then it can give an upward bias to the data,
so scientists sometimes adopt a slightly different convention:

NO-BIAS ROUNDING CONVENTION
for the situation when the number being rounded
is exactly halfway between the two rounding candidates
:

Examples:
Round each quantity to the tenths place:

[beautiful math coming... please be patient] $2.35\,$ rounds to $\,2.4$ (notice that $\,2.35\,$ is exactly halfway between $\,2.3\,$ and $\,2.4\,$, and the digit "$\,3\,$" is odd)
[beautiful math coming... please be patient] $2.65\,$ rounds to $\,2.6$ (notice that $\,2.65\,$ is exactly halfway between $\,2.6\,$ and $\,2.7\,$, and the digit "$\,6\,$" is even)
[beautiful math coming... please be patient] $2.8501\,$ rounds to $\,2.9\,$ (notice that $\,2.8501\,$ is closer to $\,2.9\,$ than $\,2.8\,$; the special rule does not apply here)

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Rewriting Fractions as a Whole Number plus a Fraction

 
 
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
(MAX is 23; there are 23 different problem types.)