Converting from big units to smaller units

You can not make a measurement more precise by converting it to a different unit. The precision was determined when the measurement was made. (the big sort-of exception)

For example, suppose you measure a path. You tell someone it is 58 feet long. Knowing you are a truthful person, he or she will think the length of the path is closer to 58 feet than it is to 57 feet or 59 feet.

Suppose we convert this length to centimeters. By definition, 1 foot = 30.48 centimeters.

58 feet × 30.48 = 1767.84 centimeters.

But wait a minute! How big is 0.04 centimeters? It is about the width of the thinnest leads for mechanical pencils. The vertical lines in the type you are reading are probably about 0.04 cm thick. Did you measure the path with that degree of precision? Of course not. The “4” at the end of “1767.84” is meaningless. It is just garbage left over from doing multiplication.

How much of the “1767.84” is meaningful?

The standard assumption is that the “real” length of the path must be greater than halfway between 57 and 58 feet, because if it were less you would have said “57.” And it must be less than halfway between 58 and 59, because if it were any greater you would have said “59.”

Half a foot is, by definition, 15.24 centimeters. If we only know the length of the path plus or minus 15 centimeters, surely the .8 cm in our answer isn't meaningful. Nor is the 7 cm. We will be more truthful if we round up to the nearest value in the tens column.

Your measurement of 58 feet converts to 1770 centimeters, not 1767.84.

Suppose the original measurement were more precise. You can show that by adding decimal places to your measurement. Suppose you use a tape marked off in tenths of a inch and get a length of 58.1 feet for the path. This measurement says the “real” length lies between 58.05 and 58.15 feet. 0.05 feet (half of a tenth of a foot) is 1.524 centimeters. We now know the length plus or minus about 1.5 cm. That's a lot less than 10, so we can be quite sure that the ten's place ought to be a “6”. rounding off to the units place.

A measurement of 58.1 feet converts to 1768 centimeters.

Now we seem to have a paradox: the measurement got bigger (58.1 feet instead of 58 feet), but the conversion got smaller (1768 cm instead of 1770 cm). The problem comes from not knowing what the “0” in “1770 cm” meant. Did it mean, we know this amount to the nearest centimeter, or did it just mean, we don't know what belongs in this place. (In our example, that is what it meant.) Scientists avoid this kind of problem by using scientific notation for measurements and by stating the margin of error.

Experiment with the converter and see how the answer changes as decimal places with zeroes are added to the input.

Many people are mystified by the results from a calculator that takes account of the presumed preciseness of the measurement. To avoid this, we may meddle with your input, by adding enough zeroes to get a significant digit in one's place in the answer.

NOTE

If you are trying to use the converter to get a conversion factor, by entering “1” in the input box for the first unit, you will not get a useful answer. Try entering “1.000000000000”, or better, use the value in the web page on that unit.

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