1.11: Scientific Notation

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Powers of Ten

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

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

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

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 $5\times10,000$ or $5\times10^4$.^[1]

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

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 $a\times10^n$, where $n$ is an integer and $a$ is any real number such that $1\leq{a}<10$.

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

Exercise $\PageIndex{1}$

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?

Answer

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

Earth’s mass is larger because it’s a $25$-digit number and Mars’ mass is a $24$-digit number, but it might take a lot of work counting the zeros to be sure.

Exercise $\PageIndex{2}$

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

Answer: Earth’s mass is about ten times larger, because the power of $10$ is $1$ higher than that of Mars.

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 of magnitude; 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.

Exercise $\PageIndex{3}$

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?

Answer: A chlorine atom’s radius is larger because it has $9$ zeros before the significant digits begin, but a hydrogen atom’s radius has $10$ zeros before the significant digits begin. As above, counting the zeros is a pain in the neck.

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

Exercise $\PageIndex{4}$

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

Answer: The chlorine atom has a larger radius because its power of $10$ is $1$ higher than that of the hydrogen atom. (Remember that $-10$ is larger than $-11$ because $-10$ is farther to the right on a number line.)

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 $\PageIndex{5}$

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. $3.5\times10^4$

10. $9.012\times10^7$

11. $8.25\times10^{-3}$

12. $1.4\times10^{-5}$

Answer

5. $1.234 \times 10^3$

6. $1.02 \times 10^7$

7. $8.7 \times 10^{-4}$

8. $7.32 \times 10^{-2}$

9. $35,000$

10. $90,120,000$

11. $0.00825$

12. $0.000014$

You may be familiar with a shortcut for multiplying numbers with zeros on the end; for example, to multiply $300\times4,000$, we can multiply the significant digits $3\times4=12$ 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: $10^m\cdot10^n=10^{m+n}$

Exercises $\PageIndex{6}$

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

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

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

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

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

Answer

13. $8 \times 10^7$

14. $3.5 \times 10^{13}$

15. $6 \times 10^{-5}$

16. $4.8 \times 10^5$

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 $\PageIndex{7}$

The mass of a proton is $1.67\times10^{-27}$ kg. The mass of an electron is $9.11\times10^{-31}$ 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 $10^9$.)

Answer

17. the proton’s mass is roughly $1,830$ or $1.83 \times 10^3$ times larger

18. $1.67 \times 10^{-21}$ kg

19. $1.67 \times 10^{-18}$ kg

Engineering Notation

Closely related to scientific notation is engineering notation, which uses only multiples of $1,000$. This is the way large numbers are often reported in the news; if roughly $37,000$ people live in Oregon City, we say “thirty-seven thousand” and we might see it written as “37 thousand”; it would be unusual to think of it as $3.7\times10,000$ and report the number as “three point seven ten thousands”.

One thousand = $10^3$, one million = $10^6$, one billion = $10^9$, one trillion = $10^{12}$, and so on.

In engineering notation, the power of $10$ is always a multiple of $3$, and the other part of the number must be between $1$ and $1,000$.

A number is written in engineering notation if it is written in the form $a\times10^n$, where $n$ is a multiple of $3$ and $a$ is any real number such that $1\leq{a}<1,000$.

Note: Prefixes for large numbers such as kilo, mega, giga, and tera are essentially engineering notation, as are prefixes for small numbers such as micro, nano, and pico. We’ll see these in another module.

Exercises $\PageIndex{8}$

Write each number in engineering notation, then in scientific notation.

20. The U.S. population is around $330.2$ million people.^[2]

21. The world population is around $7.68$ billion people.^[3]

22. The U.S. national debt is around $26.6$ trillion dollars.^[4]

Answer

20. $330.2 \times 10^6$; $3.302 \times 10^8$

21. $7.68 \times 10^9$; $7.68 \times 10^9$ \

22. (26.6 \times 10^{12}\); $2.66 \times 10^{13}$

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 fiscaldata.treasury.gov/datasets/debt-to-the-penny/debt-to-the-penny↵

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

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Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

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