David O. answered • 07/14/21
Math and Computer Science Tutoring
Hello, Alyssa!
When we are discussing the nature of decimal numbers, "precision" is defined as the amount of numbers that come after the decimal point. They have similar names to the numbers that come before it. For example, in the number 123.45, we can see that the number 4 is in the "tenths" place; the number 5 is in the hundredths. As such, we would say that 123.45 has precision to the hundredths place. If we were to move the decimal one place to the left to produce 12.345, then it would have precision to the thousandths place, and so forth.
Since we are discussing precision, I will assume we are talking about entry-level physics - in which case, the precision of multiplication is determined by the amount of significant figures. Significant figures represent the accuracy of our measurement, determined by our measuring tool; if we seek to measure in inches using a foot-long ruler, but it only shows values for whole inches or tenths of an inch, we can only record values that match that, such as 1.0 inches, or 2.5 inches. Note that trailing zeroes are used to make sure we are clear that we are working with precision to the tenths place. To add another zero (which produces 1.00 inches) is considered dishonest as that says, implicitly, that our measuring tool could measure to the hundredths place of precision.
When multiplying, the precision is determined by the least precise number. If we multiply two numbers together, we must round our result to the place of the final significant figure of the least precise number. Consider the following equation...
2.5 * 3.7
When we move to calculate this, we would receive the answer of 9.25. However, this does not match the precision, and is thus not entirely correct. We can see that both of our numbers have only have precision to the tenths place; we shouldn't produce a number with precision to the hundredths, as that implies that both numbers were at least accurate to the hundredths place. So, we round the 9.25 to 9.3, thereby making it match the precision of our measurements.
Let's try an example. Consider the equation...
10.005 * 12.10
10.005 is accurate to the thousandths place, as its final digit is 5, or five thousandths. 12.10 is accurate to the hundredths place, as the 0 - the last digit - is there. We can multiply these numbers to produce 121.0605; however, this number needs to be made precise to the hundredths place, as that is the precision of the least precise number in the equation; we round it to 121.06; our multiplication is as precise as it needs to be! Success!
Now lets try a more complex example. Consider the equation...
12.0112 * 22.002 * 30.035
We can see that the number 12.0112 is precise to the ten-thousandths place, but 22.002 and 30.035 are only precise to the thousandths place; since they are the least precise, our answer is precise to the thousandths place. Feel free to calculate this on your own, and be sure to round to the thousandths place to ensure your result is accurate, and true to your measurements.
Hopefully this helps your understanding!
