ORCCA Scientific Notation

Section 6.3 Scientific Notation

Having just learned more about exponents, including negative exponents, we can discuss a format used for very large and very small numbers called scientific notation.

Subsection 6.3.1 The Basics of Scientific Notation

An October 3, 2016 CBS News headline¹ read:

http://www.cnsnews.com/news/article/terence-p-jeffrey/federal-debt-fy-2016-jumped-142282704745246

Federal Debt in FY 2016 Jumped $\$1{,}422{,}827{,}047{,}452.46$—that's $\$12{,}036$ Per Household.

The article also later states:

By the close of business on Sept. 30, 2016, the last day of fiscal 2016, it had climbed to $\$19{,}573{,}444{,}713{,}936.79\text{.}$

When presented in this format, trying to comprehend the value of these numbers can be overwhelming. More commonly, such numbers would be presented in a descriptive manner:

The federal debt climbed by $1.42$ trillion dollars in 2016.
The federal debt was $19.6$ trillion dollars at the close of business on Sept. 30, 2016.

Unless we're presented with such news items, most of us deal with numbers no larger than the thousands in our daily life. In science, government, business, and many other disciplines, it's not uncommon to deal with much larger numbers. When numbers get this large, it can be hard to distinguish between a number that has nine or twelve digits. On the other hand, we have descriptive language that allows us grasp the value and not be lost in the sheer size of the number.

We have descriptive language for all numbers, based on the place value of the different digits: ones, tens, thousands, ten thousands, etc. We tend to rely upon this language more when we start dealing with larger numbers. Here's a chart for some of the most common numbers we see and use in the world around us:

Table 6.3.1. Whole Number Powers of $10$

Number	US English Name	Power of $10$
$1$	one	$10^{0}$
$10$	ten	$10^{1}$
$100$	hundred	$10^{2}$
$1{,}000$	one thousand	$10^{3}$
$10{,}000$	ten thousand	$10^{4}$
$100{,}000$	one hundred thousand	$10^{5}$
$1{,}000{,}000$	one million	$10^{6}$
$1{,}000{,}000{,}000$	one billion	$10^{9}$

Each number above has a corresponding power of ten and this power of ten will be important as we start to work with the content in this section.

This descriptive language also covers even larger numbers: trillion, quadrillion, quintillion, sextillion, septillion, and so on. There's also corresponding language to describe very small numbers, such as thousandth, millionth, billionth, trillionth, etc.

Through centuries of scientific progress, humanity became increasingly aware of very large numbers and very small measurements. As one example, the star that is nearest to our sun is Proxima Centauri ². Proxima Centauri is about $25{,}000{,}000{,}000{,}000$ miles from our sun. Again, many will find the descriptive language easier to digest: Proxima Centauri is about $25$ trillion miles from our sun.

imagine.gsfc.nasa.gov/features/cosmic/nearest_star_info.html

To make computations involving such numbers more manageable, a standardized notation called scientific notation was established. The foundation of scientific notation is the fact that multiplying or dividing by a power of $10$ will move the decimal point of a number so many places to the right or left, respectively.

Checkpoint 6.3.2.

Multiplying a number by $10^n$ where $n$ is a positive integer had the effect of moving the decimal point $n$ places to the right.

Every number can be written as a product of a number between $1$ and $10$ and a power of $10\text{.}$ For example, $650 = 6.5 \times 100\text{.}$ Since $100 = 10^2\text{,}$ we can also write

\begin{equation*} 650 = 6.5 \times 10^{2} \end{equation*}

and this is our first example of writing a number in scientific notation.

Definition 6.3.3.

A positive number is written in scientific notation when it has the form $a \times 10^n$ where $n$ is an integer and $1 \le a \lt 10 \text{.}$ In other words, $a$ has precisely one digit to the left of the decimal place. The exponent $n$ used here is called the number's order of magnitude. The number $a$ is sometimes called the significand or the mantissa.

Some conventions do not require $a$ to be between $1$ and $10\text{,}$ excluding both values, but that is the convention used in this book.

Subsection 6.3.2 Scientific Notation for Large Numbers

To write a numbers larger than $10$ in scientific notation, we write a decimal point after the first non-zero digit of the number and then count the number of places between where the decimal point originally was and where it now is. Scientific notation communicates the size of a number and the order of magnitude just as quickly, but with no need to write long strings of zeros or to try to decipher the language of quintillions, sextillions, etc.

Example 6.3.4.

To get a sense of how scientific notation works, let's consider familiar lengths of time converted to seconds.

Length of Time	Length in Seconds	Scientific Notation
one second	1 second	$1 \times 10^{0}$ second
one minute	60 seconds	$6 \times 10^{1}$ seconds
one hour	3600 seconds	$3.6 \times 10^{3}$ seconds
one month	2,628,000 seconds	$2.628 \times 10^{6}$ seconds
ten years	315,400,000 seconds	$3.154 \times 10^{8}$ seconds
79 years (about a lifetime)	2,491,000,000 seconds	$2.491 \times 10^{9}$ seconds

Checkpoint 6.3.5.

Checkpoint 6.3.6.

Subsection 6.3.3 Scientific Notation for Small Numbers

Scientific notation can also be useful when working with numbers smaller than $1\text{.}$ As we saw in Table 6.3.1, we can denote thousands, millions, billions, trillions, etc., with positive integer exponents on $10\text{.}$ We can similarly denote numbers smaller than $1$ (which are written as tenths, hundredths, thousandths, millionths, billionths, trillionths, etc.), with negative integer exponents on $10\text{.}$ This relationship is outlined in Table 6.3.7.

Table 6.3.7. Negative Integer Powers of $10$

Number	English Name	Power of $10$
$1$	one	$10^{0}$
$0.1$	one tenth	$\frac{1}{10}=10^{-1}$
$0.01$	one hundredth	$\frac{1}{100}=10^{-2}$
$0.001$	one thousandth	$\frac{1}{1{,}000}=10^{-3}$
$0.0001$	one ten thousandth	$\frac{1}{10{,}000}=10^{-4}$
$0.00001$	one hundred thousandth	$\frac{1}{100{,}000}=10^{-5}$
$0.000001$	one millionth	$\frac{1}{1{,}000{,}000}=10^{-6}$
$0.000000001$	one billionth	$\frac{1}{1{,}000{,}000{,}000}=10^{-9}$

To see how this works with a digit other than $1\text{,}$ let's look at $0.05\text{.}$ When we state $0.05$ as a number, we say “5 hundredths.” Thus, $0.05=5\times \frac{1}{100}\text{.}$ The fraction $\frac{1}{100}$ can be written as $\frac{1}{10^2}\text{,}$ which we know is equivalent to $10^{-2}\text{.}$ Using negative exponents, we can then rewrite $0.05$ as $5\times10^{-2}\text{.}$ This is the scientific notation for $0.05\text{.}$

In practice, we won't generally do that much computation. To write a small number in scientific notation we start as we did before and place the decimal point behind the first non-zero digit. We then count the number of decimal places between where the decimal had originally been and where it now is. Keep in mind that negative powers of ten are used to help represent very small numbers (smaller than $1$) and positive powers of ten are used to represent very large numbers (larger than $1$). So to convert $0.05$ to scientific notation, we have:

\begin{equation*} 0\overbrace{.\highlight{05}}^{2\text{ places}}=5\times 10^{-2} \end{equation*}

Example 6.3.8.

In quantum mechanics, there is an important value called the Planck Constant ³. Written as a decimal, the value of the Planck constant (rounded to 4 significant digits) is

\begin{equation*} 0.0000000000000000000000000000000006626\text{.} \end{equation*}

en.wikipedia.org/wiki/Planck_constant

In scientific notation, this number will be $6.626\times 10^{\mathord{?}}\text{.}$ To determine the exponent, we need to count the number of places from where the decimal is when the number is written as

\begin{equation*} 0.0000000000000000000000000000000006626 \end{equation*}

to where it will be when written in scientific notation:

\begin{equation*} 0\overbrace{.\highlight{0000000000000000000000000000000006}}^{34\text{ places}}626 \end{equation*}

As a result, in scientific notation, the Planck Constant value is $6.626 \times 10^{-34}\text{.}$ It will be much easier to use $6.626 \times 10^{-34}$ in a calculation, and an added benefit is that scientific notation quickly communicates both the value and the order of magnitude of the Planck constant.

Checkpoint 6.3.9.

Checkpoint 6.3.10.

Checkpoint 6.3.11.

Subsection 6.3.4 Multiplying and Dividing Using Scientific Notation

One main reason for having scientific notation is to make calculations involving immensely large or small numbers easier to perform. By having the order of magnitude separated out in scientific notation, we can separate any calculation into two components.

Example 6.3.12.

On Sept. 30th, 2016, the US federal debt was about $\$19{,}600{,}000{,}000{,}000$ and the US population was about $323{,}000{,}000\text{.}$ What was the average debt per person that day?

Calculate the answer using the numbers provided, which are not in scientific notation.
First, confirm that the given values in scientific notation are $1.96 \times 10^{13}$ and $3.23 \times 10^8\text{.}$ Then calculate the answer using scientific notation.

Explanation

We've been asked to answer the same question, but to perform the calculation using two different approaches. In both cases, we'll need to divide the debt by the population.

We may need to be working a calculator to handle such large numbers and we have to be careful that we type the correct number of zeros.

\begin{gather*} \frac{19600000000000}{323000000}\approx 60681.11 \end{gather*}
To perform this calculation using scientific notation, our work would begin by setting up the quotient $\frac{1.96 \times 10^{13}}{3.23 \times 10^8}\text{.}$ Dividing this quotient follows the same process we did with variable expressions of the same format, such as $\frac{1.96 w^{13}}{3.23 w^8}\text{.}$ In both situations, we'll divide the coefficients and then use exponent rules to simplify the powers.

\begin{align*} \frac{1.96 \times 10^{13}}{3.23 \times 10^8} \amp= \frac{1.96 }{3.23} \times\frac{10^{13}}{ 10^8}\\ \amp\approx 0.6068111 \times 10^5\\ \amp\approx 60681.11 \end{align*}

The federal debt per capita in the US on September 30th, 2016 was about $\$60{,}681.11$ per person. Both calculations give us the same answer, but the calculation relying upon scientific notation has less room for error and allows us to perform the calculation as two smaller steps.

Whenever we multiply or divide numbers that are written in scientific notation, we must separate the calculation for the coefficients from the calculation for the powers of ten, just as we simplified earlier expressions using variables and the exponent rules.

Example 6.3.13.

Multiply $\left( 2\times 10^5 \right)\left( 3\times10^4 \right)\text{.}$
Divide $\dfrac{8\times 10^{17}}{4\times 10^2}\text{.}$

Explanation

We will simplify the significand/mantissa parts as one step and then simplify the powers of $10$ as a separate step.

\begin{align*} \left( 2\times 10^5 \right)\left( 3\times10^4 \right) \amp= \left( 2\times 3 \right)\times \left(10^5 \times 10^4 \right)\\ \amp= 6 \times 10^{9} \end{align*}
\begin{align*} \frac{8 \times 10^{17}}{4\times 10^2} \amp= \frac{8}{4} \times \frac{10^{17}}{10^2}\\ \amp= 2 \times 10^{15} \end{align*}

Often when we multiply or divide numbers in scientific notation, the resulting value will not be in scientific notation. Suppose we were multiplying $\left( 9.3\times 10^{17} \right)\left( 8.2 \times 10^{-6} \right)$ and need to state our answer using scientific notation. We would start as we have previously:

\begin{align*} \left( 9.3\times 10^{17} \right)\left( 8.2 \times 10^{-6} \right) \amp=\left( 9.3\times 8.2 \right)\times \left( 10^{17} \times 10^{-6} \right)\\ \amp= 76.26 \times 10^{11}\\ \end{align*}

While this is a correct value, it is not written using scientific notation. One way to covert this answer into scientific notation is to turn just the coefficient into scientific notation and momentarily ignore the power of ten:

\begin{align*} \amp=\highlight{76.26} \times 10^{11}\\ \amp= \highlight{7.626 \times 10^1} \times 10^{11} \end{align*}

Now that the coefficient fits into the proper format, we can combine the powers of ten and have our answer written using scientific notation.

\begin{align*} \amp=7.626 \times \highlight{10^1 \times 10^{11}}\\ \amp= 7.626 \times 10^{12} \end{align*}

Example 6.3.14.

Multiply or divide as indicated. Write your answer using scientific notation.

$\displaystyle \left( 8 \times 10^{21} \right)\left( 2 \times 10^{-7} \right)$
$\displaystyle \dfrac{ 2 \times 10^{-6} }{ 8 \times 10^{-19} }$

Explanation

Again, we'll separate out the work for the significand/mantissa from the work for the powers of ten. If the resulting coefficient is not between $1$ and $10\text{,}$ we'll need to adjust that coefficient to put it into scientific notation.

\begin{align*} \left( 8 \times 10^{21} \right)\left( 2 \times 10^{-7} \right) \amp= \left( 8 \times 2 \right)\times\left( 10^{21} \times 10^{-7} \right)\\ \amp= \highlight{16} \times 10^{14}\\ \amp= \highlight{1.6\times 10^1} \times 10^{14}\\ \amp= 1.6 \times 10^{15} \end{align*}

We need to remember to apply the product rule for exponents to the powers of ten.
\begin{align*} \frac{ 2 \times 10^{-6} }{ 8 \times 10^{-19} } \amp= \frac{ 2 }{ 8 }\times\frac{ 10^{-6} }{ 10^{-19} }\\ \amp= \highlight{0.25} \times 10^{13}\\ \amp= \highlight{2.5\times 10^{-1}} \times 10^{13}\\ \amp= 2.5 \times 10^{12} \end{align*}

There are times where we will have to raise numbers written in scientific notation to a power. For example, suppose we have to find the area of a square whose radius is $3\times 10^7$ feet. To perform this calculation, we first remember the formula for the area of a square, $A=s^2$ and then substitute $3\times 10^7$ for $s\text{:}$ $A = \left( 3\times 10^7 \right)^2\text{.}$ To perform this calculation, we'll need to remember to use the product to a power rule and the power to a power rule:

\begin{align*} A \amp= \left( 3\times 10^7 \right)^2\\ \amp= \left( 3\right)^2 \times \left(10^7 \right)^2\\ \amp= 9 \times 10^{14} \end{align*}

Exercises 6.3.5 Exercises

Converting To and From Scientific Notation

1.

Write the following number in scientific notation.

$910000=$

2.

Write the following number in scientific notation.

$11000=$

3.

Write the following number in scientific notation.

$2100=$

4.

Write the following number in scientific notation.

$310000=$

5.

Write the following number in scientific notation.

$0.0041=$

6.

Write the following number in scientific notation.

$0.00051=$

7.

Write the following number in scientific notation.

$0.061=$

8.

Write the following number in scientific notation.

$0.0071=$

9.

Write the following number in decimal notation without using exponents.

$8.1\times 10^{2}=$

10.

Write the following number in decimal notation without using exponents.

$9.1\times 10^{5}=$

11.

Write the following number in decimal notation without using exponents.

$1.11\times 10^{3}=$

12.

Write the following number in decimal notation without using exponents.

$2.11\times 10^{2}=$

13.

Write the following number in decimal notation without using exponents.

$3.1\times 10^{0}=$

14.

Write the following number in decimal notation without using exponents.

$4.1\times 10^{0}=$

15.

Write the following number in decimal notation without using exponents.

$5.1\times 10^{-4}=$

16.

Write the following number in decimal notation without using exponents.

$6.1\times 10^{-2}=$

17.

Write the following number in decimal notation without using exponents.

$7.09\times 10^{-3}=$

18.

Write the following number in decimal notation without using exponents.

$8.09\times 10^{-4}=$

Arithmetic with Scientific Notation

19.

Multiply the following numbers, writing your answer in scientific notation.

$(9\times 10^{4})(7\times 10^{4})=$

20.

Multiply the following numbers, writing your answer in scientific notation.

$(2\times 10^{2})(4\times 10^{3})=$

21.

Multiply the following numbers, writing your answer in scientific notation.

$(2\times 10^{4})(2\times 10^{2})=$

22.

Multiply the following numbers, writing your answer in scientific notation.

$(3\times 10^{5})(7\times 10^{5})=$

23.

Multiply the following numbers, writing your answer in scientific notation.

$(4\times 10^{3})(4\times 10^{4})=$