The conversion to and from scientific notation follows from the fact that multiplying and dividing by 10 simply moves the decimal point to the left or to the right. However, we will want to avoid using the division symbol, so we will write division using negative exponents.

When converting a number from scientific notation to standard notation, the size of the exponent tells you how many places to move the decimal and the sign (positive or negative) of the exponent tells you which direction you need to move the decimal point. Positive exponents move the decimal to the right (it makes the number bigger) and negative exponents move the decimal to the left (it makes the number smaller). When you do this, you may need to insert zeros between the decimal point and the digits you started with as placeholders. We’re using the fact that exponents represent repeated multiplication, so that (for example) multiplying by \(10^3\) means multiplying by 10 three times, which would move the decimal three spaces to the right (one for each multiplication by 10).

When converting a number from standard notation to scientific notation, we will follow a mechanical process. To get the coefficient, think of the given number as string of digits without a decimal point, and put the decimal point to the right of the leading digit (the left-most non-zero digit) and drop all the unnecessary zeros. Then to determine the exponent, think about where the decimal point you created would need to go to get back the original number, and use the corresponding value.

To work with scientific notation, you need to understand a few basic concepts about exponents. The first is that an exponent represents repeated multiplication. For example, \(10^3 = 10 \cdot 10 \cdot 10\text{.}\) The second is that negative exponents represent a reciprocal, so that \(10^{-3} = \frac{1}{10^3} = \frac{1}{10 \cdot 10 \cdot 10}\text{.}\) Also notice that this works when the negative exponent is in the denominator, so that \(\frac{1}{10^{-3}} = 10^3\text{.}\) Lastly, when you have a fraction involving exponents on the top and bottom of the fraction, you can cancel out pairs of terms (one on top and one on bottom).

There are some formulas that you can use for this, but the formulas tend to obscure the ideas, rather than adding clarity. So you will need to go out on the internet if you want to find and use the formulas. The examples will show you the logic, and if you understand the logic then you won’t need the formulas.

Some calculators are capable of handling scientific notation. You would want to look for a button that has an EE or EXP on it, and you would want to find a tutorial (or experiment with the calculator) to make sure that you understand how to use it properly. But while you could use such a calculator to do your calculations, it is still important to understand the basic algebra that goes on behind it.

The key idea that we use is that when we have products and quotients (but not sums and differences), we can do the arithmetic in any order we want and still get the same answer. (Technically the last fact is a combination of two algebraic properties known as the commutative property and the associative property of multiplication.) The procedure we will follow is to multiply and/or divide all of the coefficients and all of the exponent terms separately, and then combine the results in the end.

This is a good time to raise a point about significant figures. A system of significant figures represents a method for tracking the level of meaningful accuracy of the digits in a calculation. One of the great challenges in empirical work is that measurements are inherently imprecise. (We’ll talk a little more about this in a later section.) If your digital scale might read 13 grams, but you won’t really know whether the weight is closer to 12.6 grams or 13.4 grams. And so the 13 grams is actually just an estimate. When multiple values in a calculation are estimates, it can often be useful to keep track of the level of the estimate, and this is done through significant figures.

The general idea is that you can track significant figures in scientific notation by counting the number of decimal points there are in the representations and doing some bookkeeping with those values. For our purposes, we’re going to skip this and just generally aim to keep at least 4 decimals. Although there are some useful ideas that can be developed in terms of really thinking through the concept of computational errors, it’s a little bit off target for the big picture goals that we’re looking for. (If you wish to pursue this further, there is a link below.)