Understanding Significant Figures in Measurements

♪♪ (whirring) Measuring tools don't always give you

the same answer every time. The answers they give you aren't always correct, and even if they are,

sometimes we don't need to keep every digit. We'll find out more about how accurate we need to be in this segment of "Physics in Motion"

on significant figures. For instance, when you measure a circle, you find the distance around a circle, its circumference,

by multiplying the diameter by the constant pi. There's just one little issue here. Pi goes on forever.

It never stops. It never repeats. You could keep writing your circle of circumference forever, until your hand fell off,

but how exact does this measurement really need to be? Can you just write down three and four or five more numbers

and call it a day? When do you stop? Does it matter? As you may have guessed, yes, it matters,

especially in physics. Scientists have established a way of dealing with how precise a measurement is using significant figures.

They're the important digits in a number based on how exact you are when you report a number. Our instruments and tools are only exact to a point.

They have limits. In order to express how exact we are being in our measurements, we need to decide which digits or figures are important

or significant. That's what significant figures are. We often call them sig figs for short.

Here are some of the rules we follow to do that. First of all, all non-zero digits are always counted as significant figures.

Any zeroes between the non-zero digits are always significant. So, that means these, the highlighted zeroes are sig figs.

That's the two zeroes in one-zero-zero-four, the zero in between four and nine, and final zeroes in the decimal part

of a number are also significant. These zeroes are important because they show how exact our measurements are.

For example, 10.50 meters says measured to the nearest 0.01 meter, and we verified that the object was not 10.51 meters

or 10.49 meters long. But zeroes between the decimal point and non-zero digits to the right of the decimal point

are not significant. For example, 0.00060450 has five significant figures. All those other zeroes are not significant.

0.0100 has three sig figs, and the other two zeroes are not, and 0.00006010 has four.

The other five zeroes, again, are not sig figs. And final zeroes in a whole number are also not significant.

1020 has three significant figures, 123000 has three significant figures, and 1203400 has five.

Another rule to remember-- when you're doing addition and subtraction calculations, you limit the number of sig figs in your final answer

to the number of decimal places in your least exact measurement. If your least exact measurement is a whole number,

your final answer must be a whole number, too. Or, if your least exact measurement is to three decimal places,

your final answer should also be to three decimal places. Let's do some measurements,

and we'll see where that rule comes into play. Let's say I measured these three pieces of wood. This one is 3.141 centimeters,

this one is 2.72 centimeters, and this one is 14 centimeters. So, let's add those three up,

and we get 19.861, and when adding or subtracting sig figs, I round to the least exact measurement, which is 14,

so I can have only one digit to the right of the decimal. However, 19.861 is closer to 19.9 than 19.8, so I will round up.

The correct answer is 19.9 centimeters. My total can only have one digit to the right of my decimal point because of the 14.0.

So, it would be 19.861, but we only want three sig figs, so our final answer is 19.9.

One piece of advice when you're dealing with sig figs-- if you're doing a calculation with several steps, keep at least one more digit than you need

until the last step. We do that because when we round numbers, we lose some information.

If we keep the digits 'til the end, we can make sure we're not losing more information than we want to before we get to the final answer.

When you do multiplication and division, it's a slightly different rule from addition and subtraction where you have to line up the decimal points.

So, for multiplication and division, the answer must have the same number of sig figs as the least exact measurement.

Let's see how that works. Suppose we are measuring the top of this piece of wood, and we want to determine the area of the surface.

The width has four significant digits, 7.344 centimeters, and the length only has two, 8.8 centimeters.

That's 64.6272 centimeters squared. Remember, when multiplying or dividing, the answer must have the same number of sig figs

as the least exact measurement. So, how many should I have? If you said two for the 8.8 centimeters measured,

you got it. So, the answer of 64.6272 should only have 2 sig figs, and that is-- do you know?

Sixty-five square centimeters. We round up from the 64.6 to 65 square centimeters. So, that's the deal with sig figs,

which show us the degree of accuracy in a measurement. That's it for this segment of "Physics in Motion." We'll see ya next time.

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