Module 11: Scientific Notation — Technical Mathematics

sboschman

17 Module 11: Scientific Notation — Technical Mathematics

Scientific Notation

Powers of Ten

Decimal notation is based on powers of $10$ : $0.1$ is $frac{1}{10^1}$ , $0.01$ is $frac{1}{10^2}$ , $0.001$ is $frac{1}{10^3}$ , and so on.

We represent these powers with negative exponents: $frac{1}{10^1}=10^{-1}$ , $frac{1}{10^2}=10^{-2}$ , $frac{1}{10^3}=10^{-3}$ , etc.

Negative exponents: $frac{1}{10^n}=10^{-n}$

Note: This is true for any base, not just $10$ , but we will focus only on $10$ in this course.

With our base $10$ number system, any power of $10$ can be written as a $1$ in a certain decimal place.

$10^{4}$	$10^{3}$	$10^{2}$	$10^{1}$	$10^{0}$	$10^{-1}$	$10^{-2}$	$10^{-3}$	$10^{-4}$
$10,000$	$1,000$	$100$	$10$	$1$	$0.1$	$0.01$	$0.001$	$0.0001$

If you haven’t watched the video “Powers of Ten” from 1977 on YouTube, take ten minutes right now and check it out. Your mind will never be the same again.

Scientific Notation

Let’s consider how we could rewrite some different numbers using these powers of $10$ .

Let’s take $50,000$ as an example. $50,000$ is equal to $5times10,000$ or $5times10^4$ .^[1]

Looking in the other direction, a decimal such as $0.0007$ is equal to $7times0.0001$ or $7times10^{-4}$ .

The idea behind scientific notation is that we can represent very large or very small numbers in a more compact format: a number between $1$ and $10$ , multiplied by a power of $10$ .

A number is written in scientific notation if it is written in the form $atimes10^n$ , where $n$ is an integer and $a$ is any real number such that <img src=”https://ecampusontario.pressbooks.pub/app/uploads/sites/2699/2022/06/quicklatex.com-d38db1d8215c5f98a4d3118ba73057cd_l3.png” class=”ql-img-inline-formula quicklatex-auto-format” alt=”1leq{a}.

Note: An integer is a number with no fraction or decimal part: … $-3$ , $-2$ , $-1$ , $0$ , $1$ , $2$ , $3$ …

Exercises

1. The mass of the Earth is approximately $5,970,000,000,000,000,000,000,000$ kilograms. The mass of Mars is approximately $639,000,000,000,000,000,000,000$ kilograms. Can you determine which mass is larger?

Clearly, it is difficult to keep track of all those zeros. Let’s rewrite those huge numbers using scientific notation.

Exercises

2. The mass of the Earth is approximately $5.97times10^{24}$ kilograms. The mass of Mars is approximately $6.39times10^{23}$ kilograms. Can you determine which mass is larger?

It is much easier to compare the powers of $10$ and determine that the mass of the Earth is larger because it has a larger power of $10$ . You may be familiar with the term order ofmagnitude; this simply refers to the difference in the powers of $10$ of the two numbers. Earth’s mass is one order of magnitude larger because $24$ is $1$ more than $23$ .

We can apply scientific notation to small decimals as well.

Exercises

3. The radius of a hydrogen atom is approximately $0.000000000053$ meters. The radius of a chlorine atom is approximately $0.00000000018$ meters. Can you determine which radius is larger?

Again, keeping track of all those zeros is a chore. Let’s rewrite those decimal numbers using scientific notation.

Exercises

4. The radius of a hydrogen atom is approximately $5.3times10^{-11}$ meters. The radius of a chlorine atom is approximately $1.8times10^{-10}$ meters. Can you determine which radius is larger?

The radius of the chlorine atom is larger because it has a larger power of $10$ ; the digits $1$ and $8$ for chlorine begin in the tenth decimal place, but the digits $5$ and $3$ for hydrogen begin in the eleventh decimal place.

Scientific notation is very helpful for really large numbers, like the mass of a planet, or really small numbers, like the radius of an atom. It allows us to do calculations or compare numbers without going cross-eyed counting all those zeros.

Exercises

Write each of the following numbers in scientific notation.

5.
$1,234$

6.
$10,200,000$

7.
$0.00087$

8.
$0.0732$

Convert the following numbers from scientific notation to standard decimal notation.

9.
$3.5times10^4$

10.
$9.012times10^7$

11.
$8.25times10^{-3}$

12.
$1.4times10^{-5}$

You may be familiar with a shortcut for multiplying numbers with zeros on the end; for example, to multiply $300times4,000$ , we can multiply the significant digits $3times4=12$ and count up the total number of zeros, which is five, and write five zeros on the back end of the $12$ : $1,200,000$ . This shortcut can be applied to numbers in scientific notation.

To multiply powers of $10$ , add the exponents: $10^mcdot10^n=10^{m+n}$

Exercises

Multiply each of the following and write the answer in scientific notation.

13.
$(2times10^3)(4times10^4)$

14.
$(5times10^4)(7times10^8)$

15.
$(3times10^{-2})(2times10^{-3})$

16.
$(8times10^{-5})(6times10^9)$

When the numbers get messy, it’s probably a good idea to use a calculator. If you are dividing numbers in scientific notation with a calculator, you may need to use parentheses carefully.

Exercises

The mass of a proton is $1.67times10^{-27}$ kg. The mass of an electron is $9.11times10^{-31}$ kg.

17. Divide these numbers using a calculator to determine approximately how many times greater the mass of a proton is than the mass of an electron.

18. What is the approximate mass of one million protons? (Note: one million is $10^6$ .)

19. What is the approximate mass of one billion protons? (Note: one billion is $10^9$ .)

For some reason, although we generally try to avoid using the “x” shaped multiplication symbol, it is frequently used with scientific notation. ↵
August 27, 2020 estimate from https://www.census.gov/popclock/↵
August 27, 2020 estimate from https://www.census.gov/popclock/↵
August 27, 2020 data from https://fiscaldata.treasury.gov/datasets/debt-to-the-penny/debt-to-the-penny↵

17 Module 11: Scientific Notation — Technical Mathematics

Scientific Notation

Powers of Ten

Scientific Notation

License

Share This Book