### Scientific Notation

Consider the number six hundred and forty three thousand, two hundred and ninety seven. It would be written as 643,297, with each digit’s position having a “value” assigned to it. You may have seen a table like this before:

\begin{align*}& \text{hundred-thousands} \quad \text{ten-thousands} \quad \text{thousands} \quad \text{hundreds} \quad \text{tens} \quad \text{units}\\ & \qquad \quad \ 6 \qquad \qquad \qquad 4 \qquad \qquad \quad 3 \qquad \qquad 2 \qquad \quad 9 \qquad \ 7\end{align*}

We’ve seen that when we write an exponent above a number, it means that we have to multiply a certain number of copies of that number together. You may also know already that any number with a zero exponent equals 1, and negative exponents represent fractional values.

Look carefully at the table above. Do you notice that all the column headings are powers of ten? Here they are listed from greatest to least:

\begin{align*}\text{Hundred thousands}& & {100, \!000} &= 10^5\\ \text{Ten thousands}& &{10, \!000} &= 10^4\\ \text{Thousands}& &{1, \!000} &= 10^3\\ \text{Hundreds}& &{100} &= 10^2\\ \text{Tens}& &{10} &= 10^1\end{align*}

Even the “units” column is really just a power of ten. ** Unit** means 1, and 1 is \begin{align*}10^0.\end{align*}

If we divide 643,297 by 100,000 we get 6.43297; if we multiply 6.43297 by 100,000 we get 643, 297. But we have just seen that 100,000 is the same as \begin{align*}10^5\end{align*}, so if we multiply 6.43297 by \begin{align*}10^5\end{align*} we should also get 643,297. In other words,

\begin{align*}643,297 = 6.43297 \times 10^5\end{align*}

**Writing Numbers in Scientific Notation**

In scientific notation, numbers are always written in the form \begin{align*}a \times 10^b\end{align*} , where \begin{align*}b\end{align*} is an integer and \begin{align*}a\end{align*} is between 1 and 10 (meaning it has exactly 1 nonzero digit before the decimal). This notation is especially useful for numbers that are either very small or very large.

Here’s a set of examples:

\begin{align*}1.07 \times 10^4 &= 10, \!700\\ 1.07 \times 10^3 &= 1, \!070\\ 1.07 \times 10^2 &= 107\\ 1.07 \times 10^1 &= 10.7\\ 1.07 \times 10^0 &= 1.07\\ 1.07 \times 10^{\text{-}1} &= 0.107\\ 1.07 \times 10^{\text{-}2} &= 0.0107\\ 1.07 \times 10^{\text{-}3} &= 0.00107\\ 1.07 \times 10^{\text{-}4} &= 0.000107\end{align*}

Look at the first example and notice where the decimal point is in both expressions.

So the exponent on the ten acts to move the decimal point over to the right. An exponent of 4 moves it 4 places and an exponent of 3 would move it 3 places.

This makes sense because each time you multiply by 10, you move the decimal point one place to the right. 1.07 times 10 is 10.7, then 10.7 times 10 again is 107.0, and so on.

Similarly, if you look at the later examples in the table, you can see that a negative exponent on the 10 means the decimal point moves that many places to the left. This is because multiplying by \begin{align*}10^{\text{-}1}\end{align*} is the same as multiplying by \begin{align*}\frac{1}{10},\end{align*} which is like dividing by 10. So instead of moving the decimal point one place to the right for every multiple of 10, we move it one place to the left for every multiple of \begin{align*}\frac{1}{10}.\end{align*}

That’s how to convert numbers from scientific notation to standard form. When we’re converting numbers *to* scientific notation, however, we have to apply the whole process backwards. First we move the decimal point until it’s immediately after the first nonzero digit; then we count how many places we moved it. If we moved the decimal to the *left,* the exponent on the 10 is positive; if we moved it to the *right,* the exponent is negative.

For example, to write 0.000032 in scientific notation, we’d first move the decimal five places to the right to get 3.2; then, since we moved it right, the exponent on the 10 should be *negative* five, so 0.000032 in scientific notation is \begin{align*}3.2 \times 10^{\text{-}5}.\end{align*}

You can double-check whether you’ve got the right direction by comparing the number in scientific notation with the number in standard form, and thinking “Does this represent a *big* number or a *small* number?” A positive exponent on the 10 represents a number bigger than 10 and a negative exponent represents a number smaller than 10, and you can easily tell if the number in standard form is bigger or smaller than 10 just by looking at it.

#### Writing Numbers in Scientific Notation

Write the following numbers in scientific notation.

a) 63

\begin{align*}63 = 6.3 \times 10 = 6.3 \times 10^1\end{align*}

b) 9,654

\begin{align*}9,654 = 9.654 \times 1,000 = 9.654 \times 10^3\end{align*}

c) 653,937,000

\begin{align*}653, \!937, \!000 = 6.53937000 \times 100, \!000, \!000 = 6.53937 \times 10^8\end{align*}

d) 0.003

\begin{align*}0.003 = 3 \times \frac{1}{1000} = 3 \times 10^{\text{-}3}\end{align*}

e) 0.000056

\begin{align*}0.000056 = 5.6 \times \frac{1}{100, \!000} = 5.6 \times 10^{\text{-}5}\end{align*}

f) 0.00005007

\begin{align*}0.00005007 = 5.007 \times \frac{1}{100, \!000} = 5.007 \times 10^{\text{-}5}\end{align*}

#### Evaluating Expressions

The key to evaluating expressions involving scientific notation is to group the powers of 10 together and deal with them separately.

a) \begin{align*}(3.2 \times 10^6) \cdot (8.7 \times 10^{11})\end{align*}

\begin{align*}(3.2 \times 10^6) (8.7 \times 10^{11}) = \underbrace{3.2 \times 8.7}_{27.84} \times \underbrace{10^6 \times 10^{11}}_{10^{17}} = 27.84 \times 10^{17}.\end{align*} But \begin{align*}27.84 \times 10^{17}\end{align*} isn’t in proper scientific notation, because it has more than one digit before the decimal point. We need to move the decimal point one more place to the left and add 1 to the exponent, which gives us \begin{align*}2.784 \times 10^{18}.\end{align*}

b) \begin{align*}(5.2 \times 10^{\text{-}4}) \cdot (3.8 \times 10^{\text{-}19})\end{align*}

\begin{align*}(5.2 \times 10^{\text{-}4}) (3.8 \times 10^{\text{-}19}) &= \underbrace{5.2 \times 3.8}_{19.76} \times \underbrace{10^{\text{-}4} \times 10^{\text{-}19}}_{10^{\text{-}23}} \\ &= 19.76 \times 10^{\text{-}23} \\ &= 1.976 \times 10^{\text{-}22}\end{align*}

c) \begin{align*}(1.7 \times 10^6) \cdot (2.7 \times 10^{\text{-}11})\end{align*}

\begin{align*}(1.7 \times 10^6) (2.7 \times 10^{\text{-}11}) = \underbrace{1.7 \times 2.7}_{4.59} \times \underbrace{10^6 \times 10^{\text{-}11}}_{10^{\text{-}5}} = 4.59 \times 10^{\text{-}5}\end{align*}

When we use scientific notation in the real world, we often round off our calculations. Since we’re often dealing with very big or very small numbers, it can be easier to round off so that we don’t have to keep track of as many digits—and scientific notation helps us with that by saving us from writing out all the extra zeros. For example, if we round off 4,227,457,903 to 4,200,000,000, we can then write it in scientific notation as simply \begin{align*}4.2 \times 10^9.\end{align*}

When rounding, we often talk of **significant figures** or **significant digits**. Significant figures include

- all nonzero digits
- all zeros that come
*before*a nonzero digit and*after*either a decimal point or another nonzero digit

For example, the number 4000 has one significant digit; the zeros don’t count because there’s no nonzero digit after them. But the number 4000.5 has five significawnt digits: the 4, the 5, and all the zeros in between. And the number 0.003 has three significant digits: the 3 and the two zeros that come between the 3 and the decimal point.

#### Rounding to the Correct Amount of Significant Figures

Evaluate the following expressions. Round to 3 significant figures and write your answer in scientific notation.

a) \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})\end{align*}

b) \begin{align*}(5.2 \times 10^{-4}) \div (3.8 \times 10^{-19})\end{align*}

**Solution**

It’s easier if we convert to fractions and THEN separate out the powers of 10.

a) \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11})\end{align*} \begin{align*}(3.2 \times 10^6) \div (8.7 \times 10^{11}) &= \frac{3.2 \times 10^6}{8.7 \times 10^{11}} && \text{separate out the powers of 10}\\ & = \frac{3.2}{8.7} \times \frac{10^6}{10^{11}} && \text{evaluate each fraction (round to 3 s.f.)} \\ & = 0.368 \times 10^{(6 - 11)}\\ & = 0.368 \times 10^{\text{-}5} && \text{remember only 1 number before the decimal!}\\ & = 3.68 \times 10^{\text{-}6}\end{align*}

b) \begin{align*}(5.2 \times 10^{\text{-}4}) \div (3.8 \times 10^{\text{-}19})\end{align*} \begin{align*}(5.2 \times 10^{\text{-}4}) \div (3.8 \times 10^{\text{-}19}) & = \frac{5.2 \times 10^{\text{-}4}}{3.8 \times 10^{\text{-}19}} && \text{separate the powers of 10}\\ & = \frac{5.2}{3.8} \times \frac{10^{\text{-}4}}{10^{\text{-}19}} && \text{evaluate each fraction (round to 3 s.f.)}\\ & = 1.37 \times 10^{((\text{-}4) - (\text{-}19))}\\ & = 1.37 \times 10^{15}\end{align*}

### Example

#### Example 1

Evaluate the following expression. Round to 3 significant figures and write your answer in scientific notation.

\begin{align*}(1.7 \times 10^6) \div (2.7 \times 10^{\text{-}11})\end{align*}

\begin{align*}(1.7 \times 10^6) \div (2.7 \times 10^{\text{-}11}) & = \frac{1.7 \times 10^6}{2.7 \times 10^{\text{-}11}} && \text{next we separate the powers of 10}\\ & = \frac{1.7}{2.7} \times \frac{10^6}{10^{\text{-}11}} && \text{evaluate each fraction (round to 3 s.f.)}\\ & = 0.630 \times 10^{(6-(\text{-}11))}\\ & = 0.630 \times 10^{17}\\ & = 6.30 \times 10^{16}\end{align*}

Note that we have to leave in the final zero to indicate that the result has been rounded.

### Review

Write the numerical value of the following.

- \begin{align*}3.102 \times 10^2\end{align*}
- \begin{align*}7.4 \times 10^4\end{align*}
- \begin{align*}1.75 \times 10^{\text{-}3}\end{align*}
- \begin{align*}2.9 \times 10^{\text{-}5}\end{align*}
- \begin{align*}9.99 \times 10^{\text{-}9}\end{align*}

Write the following numbers in scientific notation.

- 120,000
- 1,765,244
- 12
- 0.00281
- 0.000000027

How many significant digits are in each of the following?

- 38553000
- 2754000.23
- 0.0000222
- 0.0002000079

Round each of the following to two significant digits.

- 3.0132
- 82.9913

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.8