Writing Scientific Notation

Learning Outcomes

Define decimal and scientific notation
Convert between scientific and decimal notation

Convert Between Scientific and Decimal Notation

Before we can convert between scientific and decimal notation, we need to know the difference between the two. Scientific notation is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. Using exponential notation, large and small numbers can be written in a way that is easier to read.

When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between $0$ and $1$ . It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.

Word	How many thousands	Number	Scientific Notation
million	$1000 x 1000$ = a thousand thousands	$1,000,000$	$10^6$
billion	$(1000 x 1000) x 1000$ = a thousand millions	$1,000,000,000$	$10^9$
trillion	$(1000 x 1000 x 1000)$ x 1000 = a thousand billions	$1,000,000,000,000$	$10^{12}$

1 billion can be written as $1,000,000,000$ or represented as $10^9$ . How would $2$ billion be represented? Since $2$ billion is $2$ times $1$ billion, then $2$ billion can be written as $2\times10^9$ .

A light year is the number of miles light travels in one year, about $5,880,000,000,000$ . That is a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number. Using scientific notation, the distance is $5.88\times10^{12}$ miles. The exponent of $12$ tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a hydrogen atom, which is about 0.00000005 mm, and in scientific notation is $5\times10^{-8}$ mm. In this case, the $-8$ tells us how many places to count to the right of the decimal.

Outlined in the box below are some important conventions of scientific notation format.

Scientific Notation

A positive number is written in scientific notation if it is written as $a\times10^{n}$ where the coefficient a is $1\leq{a}<10$ , and n is an integer.

Look at the numbers below. Which of the numbers is written in scientific notation?

Number	Scientific Notation?	Explanation
$1.85\times10^{-2}$	yes	$1\leq1.85<10$ $-2$ is an integer
$\displaystyle 1.083\times {{10}^{\frac{1}{2}}}$	no	$\displaystyle \frac{1}{2}$ is not an integer
$0.82\times10^{14}$	no	$0.82$ is not $\geq1$
$10\times10^{3}$	no	$10$ is not < $10$

Now compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.

Large Numbers		Small Numbers
Decimal Notation	Scientific Notation	Decimal Notation	Scientific Notation
$500.0$	$5\times10^{2}$	$0.05$	$5\times10^{-2}$
$80,000.0$	$8\times10^{4}$	$0.0008$	$8\times10^{-4}$
$43,000,000.0$	$4.3\times10^{7}$	$0.00000043$	$4.3\times10^{-7}$
$62,500,000,000.0$	$6.25\times10^{10}$	$0.000000000625$	$6.25\times10^{-10}$

Convert from Decimal Notation to Scientific Notation

To write a large number in scientific notation, move the decimal point to the left to obtain a number between $1$ and $10$ . Since moving the decimal point changes the value, you have to multiply the decimal by a power of $10$ so that the expression has the same value.

Let us look at an example:

$\begin{array}{r}\underset{\longleftarrow}{180,000.}=18,000.0\times10^{1}\\1,800.00\times10^{2}\\180.000\times10^{3}\\18.0000\times10^{4}\\1.80000\times10^{5}\\180,000=1.8\times10^{5}\end{array}$

Notice that the decimal point was moved $5$ places to the left, and the exponent is $5$ .

Example

Write the following numbers in scientific notation.

$920,000,000$
$10,200,000$
$100,000,000,000$

Show Solution

$\underset{\longleftarrow}{920,000,000}$ We will move the decimal point to the left. It helps to place it at the end of the number and then count how many times you move it to get one number before it that is between $1$ and $10$ . Move the decimal point $8$ times to the left and you will have $9.20000000$ . Now we can replace the zeros with an exponent of $8$ giving us $9.2\times10^{8}$ .
$\underset{\longleftarrow}{10,200,000}=10,200,000.0=1.02\times10^{7}$ . Note here how we included the $0$ and the $2$ after the decimal point. In some disciplines, you may learn about when to include both of these. Follow instructions from your teacher on rounding rules.
$\underset{\longleftarrow}{100,000,000,000}=100,000,000,000.0=1.0\times10^{11}$

To write a small number (between $0$ and $1$ ) in scientific notation, you move the decimal to the right and the exponent will have to be negative, as in the following example.

$\begin{array}{r}\underset{\longrightarrow}{0.00004}=00.0004\times10^{-1}\\000.004\times10^{-2}\\0000.04\times10^{-3}\\00000.4\times10^{-4}\\000004.\times10^{-5}\\0.00004=4\times10^{-5}\end{array}$

You may notice that the decimal point was moved five places to the right until you got to the number $4$ which is between $1$ and $10$ . The exponent is $−5$ .

Example

Write the following numbers in scientific notation.

$0.0000000000035$
$0.0000000102$
$0.00000000000000793$

Show Solution

$\underset{\longrightarrow}{0.0000000000035}=3.5\times10^{-12}$ . We moved the decimal 12 times to get to a number between $1$ and $10$
$\underset{\longrightarrow}{0.0000000102}=1.02\times10^{-8}$
$\underset{\longrightarrow}{0.00000000000000793}=7.93\times10^{-15}$

In the following video, you are provided with examples of how to convert both a large and a small number in decimal notation to scientific notation.

Convert from Scientific Notation to Decimal Notation

You can also write scientific notation as decimal notation. Recall the number of miles that light travels in a year is $5.88\times10^{12}$ , and a hydrogen atom has a diameter of $5\times10^{-8}$ mm. To write each of these numbers in decimal notation, you move the decimal point the same number of places as the exponent. If the exponent is positive, move the decimal point to the right. If the exponent is negative, move the decimal point to the left.

$\begin{array}{l}5.88\times10^{12}=\underset{\longrightarrow}{5.880000000000.}=5,880,000,000,000\\5\times10^{-8}=\underset{\longleftarrow}{0.00000005.}=0.00000005\end{array}$

For each power of $10$ , you move the decimal point one place. Be careful here and do not get carried away with the zeros—the number of zeros after the decimal point will always be $1$ less than the exponent because it takes one power of $10$ to shift that first number to the left of the decimal.

Example

Write the following in decimal notation.

$4.8\times10{-4}$
$3.08\times10^{6}$

Show Solution

$4.8\times10^{-4}$ , the exponent is negative, so we need to move the decimal to the left. $\underset{\longleftarrow}{4.8\times10^{-4}}=\underset{\longleftarrow}{.00048}$
$3.08\times10^{6}$ , the exponent is positive, so we need to move the decimal to the right. $\underset{\longrightarrow}{3.08\times10^{6}}=\underset{\longrightarrow}{3080000}$

Think About It

To help you get a sense of the relationship between the sign of the exponent and the relative size of a number written in scientific notation, answer the following questions. You can use the textbox to write your ideas before you reveal the solution.

1. You are writing a number whose absolute value is greater than 1 in scientific notation. Will your exponent be positive or negative?

2.You are writing a number whose absolute value is between 0 and 1 in scientific notation. Will your exponent be positive or negative?

3. What power do you need to put on $10$ to get a result of $1$ ?

Show Solution

You are writing a number whose absolute value is greater than 1 in scientific notation. Will your exponent be positive or negative? For numbers greater than $1$ , the exponent on $10$ will be positive when you are using scientific notation. Refer to the table presented above:

Word	How many thousands	Number	Scientific Notation
million	$1000 x 1000$ = a thousand thousands	$1,000,000$	$10^6$
billion	$(1000 x 1000) x 1000$ = a thousand millions	$1,000,000,000$	$10^9$
trillion	$(1000 x 1000 x 1000) x 1000$ = a thousand billions	$1,000,000,000,000$	$10^{12}$

2. You are writing a number whose absolute value is between $0$ and $1$ in scientific notation. Will your exponent be positive or negative? We can reason that since numbers greater than $1$ will have a positive exponent, numbers between $0$ and $1$ will have a negative exponent. Why are we specifying numbers between $0$ and $1$ ? The numbers between $0$ and $1$ represent amounts that are fractional. Recall that we defined numbers with a negative exponent as ${a}^{-n}=\frac{1}{{a}^{n}}$ , so if we have $10^{-2}$ we have $\frac{1}{10\times10}=\frac{1}{100}$ which is a number between 0 and 1.

3. What power do you need to put on $10$ to get a result of $1$ ? Recall that any number or variable with an exponent of $0$ is equal to $1$ , as in this example:

$\begin{array}{c}\frac{t^{8}}{t^{8}}=\frac{\cancel{t^{8}}}{\cancel{t^{8}}}=1\\\frac{{t}^{8}}{{t}^{8}}={t}^{8 - 8}={t}^{0}\\\text{ therefore }\\{t}^{0}=1\end{array}$

We now have described the notation necessary to write all possible numbers on the number line in scientific notation.

In the next video, you will see how to convert a number written in scientific notation into decimal notation.

Summary

Large and small numbers can be written in scientific notation to make them easier to understand. In the next section, you will see that performing mathematical operations such as multiplication and division on large and small numbers is made easier by scientific notation and the rules of exponents.

Module 7: Exponents

Learning Outcomes

Convert Between Scientific and Decimal Notation

Scientific Notation

Convert from Decimal Notation to Scientific Notation

Example

Example

Convert from Scientific Notation to Decimal Notation

Example

Think About It

Summary

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