7 x 7 = 50

If you type "7 x 7" into your calculator, you'll get 49, not 50.

But it turns out that both answers are valid, and it just depends on the context.

If you don't know already, in science and engineering we have this idea of "significant figures" or "sig figs". This idea helps keep us from thinking we're more precise than we really are.

For example, let's say you measured a table to be 5 ft long. It might not have been exactly 5 feet; it might have been 4.93 feet for example. But to the precision with which you measured, it rounds to 5 feet.

Now let's say you want to do some calculation with that number. Perhaps you also measured the width to be 3.5 feet (with greater precision) and you wanted to find the area. Well, 5 x 3.5 = 17.5, right? Wrong.

This is where significant figures come in.

What are significant figures? They're basically the digits in a given number that define how precisely it has been measured.

There are specific rules on how we determine which digits are significant and which are not.

1. If the digit is not zero, it is significant.
2. If the digit is zero, it is significant unless it is a trailing zero that comes before a decimal point.

That might be a bit confusing, so let's look at some examples:

How many sig figs are in the number 25?

Well, both digits are non-zero numbers, so they're both considered significant. Therefore, it has 2 significant figures.

How many sig figs are in the number 1.00?

We know the 1 is significant. And it turns out that the zeros are also significant because they come after the decimal point, so the answer is 3.

How many sig figs are in the number 200?

The 2 is significant, but the zeros are not, because they're trailing at the end of the number, but there's no decimal point. Therefore, the answer is 1.

How many sig figs in 101?

The zero is significant here, because it's not a trailing zero; it's between to other significant digits. There are 3 sig figs here.

Now that we know what sig figs are, we need to use them when we make calculations.

There are two main rules we use when making calculations with sig figs:

1. When multiplying and dividing, the number of sig figs in the result should be equal to the least number of sig figs that is involved in the calculation. (So if we had 5 x 3.5 again, 5 has the least number of sig figs (1), so the answer must have 1 sig fig.

2. When adding and subtracting, the precision of the result should be equal to the precision of the least precise number. (So if we had 2.0 + 3.00, the answer would be precise to the tenths place like 2: 2.0 + 3.00 = 5.0.

If the result comes out to have the wrong number of sig figs, or the wrong precision, we round it to the correct precision or the correct number of sig figs. Sometimes we may need to write it in scientific notation.

The reason we do all this is to make sure we're not stating more accuracy than we actually have. The accuracy of the result is only as good as the accuracy of its least accurate component.

Now that you know all this, you can understand why 7 x 7 = 50. The 7s each have 1 sig fig, so the answer must have 1 sig fig.

But 7.0 x 7.0? That's 49.

NOTE: This process is really only relevant to science and engineering. In regular math class, each number is considered as having infinite precision. So don't apply this to your math class!