1.7: Scientific and Engineering Notation
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Powers of Ten
Decimal notation is based on powers of 10: 0.1 is 1101, 0.01 is 1102, 0.001 is 1103, and so on.
We represent these powers with negative exponents: 1101=10−1, 1102=10−2, 1103=10−3, etc.
Negative exponents: 110n=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.
104 | 103 | 102 | 101 | 100 | 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 5×10,000 or 5×104.[1]
Looking in the other direction, a decimal such as 0.0007 is equal to 7×0.0001 or 7×10−4.
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×10n, where n is an integer and a is any real number such that 1≤a<10.
Note: An integer is a number with no fraction or decimal part: … −3, −2, −1, 0, 1, 2, 3 …
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?
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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.
2. The mass of the Earth is approximately 5.97×1024 kilograms. The mass of Mars is approximately 6.39×1023 kilograms. Can you determine which mass is larger?
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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.
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?
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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.
4. The radius of a hydrogen atom is approximately 5.3×10−11 meters. The radius of a chlorine atom is approximately 1.8×10−10 meters. Can you determine which radius is larger?
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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.
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. 3.5×104
10. 9.012×107
11. 8.25×10−3
12. 1.4×10−5
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5. 1.234×103
6. 1.02×107
7. 8.7×10−4
8. 7.32×10−2
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 300×4,000, we can multiply the significant digits 3×4=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.
Multiply each of the following and write the answer in scientific notation.
13. (2×103)(4×104)
14. (5×104)(7×108)
15. (3×10−2)(2×10−3)
16. (8×10−5)(6×109)
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13. 8×107
14. 3.5×1013
15. 6×10−5
16. 4.8×105
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.
The mass of a proton is 1.67×10−27 kg. The mass of an electron is 9.11×10−31 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 106.)
19. What is the approximate mass of one billion protons? (Note: one billion is 109.)
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17. the proton’s mass is roughly 1,830 or 1.83×103 times larger
18. 1.67×10−21 kg
19. 1.67×10−18 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 3.7×10,000 and report the number as “three point seven ten thousands”.
One thousand = 103, one million = 106, one billion = 109, one trillion = 1012, 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×10n, where n is a multiple of 3 and a is any real number such that 1≤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 next.
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]
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20. 330.2×106; 3.302×108
21. 7.68×109; 7.68×109 \
22. (26.6 \times 10^{12}\); 2.66×1013
Measurement Prefixes: Larger
Now let’s turn our attention to converting units based on their prefixes. We’ll start with large units of measure.
tera- (T) | giga- (G) | mega- (M) | kilo- (k) | [base unit] |
trillion | billion | million | thousand | one |
1,000,000,000,000 | 1,000,000,000 | 1,000,000 | 1,000 | 1 |
1012 | 109 | 106 | 103 | 100 |
Notice that the powers of these units are multiples of 3, just as with the engineering notation we saw in a previous module. Each prefix is 1,000 times the next smaller prefix, so moving one place in the chart means moving the decimal point three places. Also notice that capitalization is important; megagram (which is also called a metric ton) is Mg with a capital M, but milligram is mg with a lowercase m.
Using computer memory as an example:
1 kilobyte = 1,000 bytes
1 megabyte = 1,000 kilobytes = 1,000,000 bytes
1 gigabyte = 1,000 megabytes = 1,000,000 kilobytes, etc.
1 terabyte = 1,000 gigabytes = 1,000,000 megabytes, etc.
Note: There can be inconsistencies with different people’s understanding of these prefixes with regards to computer memory, which is counted in powers of 2, not 10. Computer engineers originally defined 1 kilobyte as 1,024 bytes because 210=1,024, which is very close to 1,000. However, we will consider these prefixes to be powers of 1,000, not 1,024. There is an explanation at https://physics.nist.gov/cuu/Units/binary.html.
23. A 514 inch floppy disk from the 1980s could store about 100 kB; a 312 inch floppy disk from the 1990s could store about 1.44 MB. By what factor was the storage capacity increased?
24. How many times greater is the storage capacity of a 2 TB hard drive than a 500 GB hard drive?
25. In an article describing small nuclear reactors that are designed to be retrofitted into coal plants, Dr. Jose Reyes of Oregon State University says “One module will produce 60 megawatts of electricity. That’s enough for about 50 thousand homes.”[5] How much electricity per home is this?
26. In the same article, Dr. Reyes says “a 60 megawatt module could produce about 60 million gallons of clean water per day using existing technologies in reverse osmosis.” What is the rate of watts per gallon?
27. The destructive power of nuclear weapons is measured in kilotons (the equivalent of 1,000 tons of TNT) or megatons (the equivalent of 1,000,000 tons of TNT). The first nuclear device ever tested, the US’s Trinity, was measured at roughly 20 kilotons on July 16, 1945. The largest thermonuclear weapon ever detonated, at 50 megatons, was the USSR’s Tsar Bomba, on October 31, 1961.[6] (Video of Tsar Bomba was declassified almost 60 years later, in August 2020.) How many times more powerful was Tsar Bomba than Trinity?
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23. the capacity increased by a factor of 14.4
24. 4 times greater
25. 1,200 megawatts per home
26. 1 watt per gallon
27. 2,500 times more powerful
Measurement Prefixes: Smaller
Now we’ll go in the other direction and look at small units of measure.
[base unit] | milli- (m) | micro- (μ or mc) | nano- (n) | pico (p) |
one | thousandth | millionth | billionth | trillionth |
1 | 0.001 | 0.000001 | 0.000000001 | 0.000000000001 |
100 | 10−3 | 10−6 | 10−9 | 10−12 |
The symbol for micro- is the Greek letter μ (pronounced “myoo”). Because this character can be difficult to replicate, you may see the letter “u” standing in for “μ” in web-based or plaintext technical articles… or you may see the prefix “mc” instead.
Again, the powers are multiples of 3; each prefix gets smaller by a factor of 11000. The negative exponents can sometime be complicated to work with, and it may help to think about things in reverse.
1 meter = 103 millimeters = 106 micrometers = 109 nanometers = 1012 picometers
1 second = 103 milliseconds = 106 microseconds = 109 nanoseconds = 1012 picoseconds
…and so on.
See https://physics.nist.gov/cuu/Units/prefixes.html for a list of more prefixes.
28. An article about network latency compares the following latency times: “So a 10 Mbps link adds 0.4 milliseconds to the RTT, a 100 Mbps link 0.04 ms and a 1 Gbps link just 4 microseconds.”[7] Rewrite these times so that they are all in terms of milliseconds, then rewrite them in terms of microseconds.
29. The wavelength of red light is around 700 nm. Infrared radiation has a wavelength of approximately 10 μm.[8] Find the ratio of these wavelengths.
30. Nuclear radiation is measured in units called Sieverts, but because this unit is too large to be practical when discussing people’s exposure to radiation, milliSieverts and microSieverts are more commonly used. In 1986, workers cleaning up the Chernobyl disaster were exposed to an estimated dose of 250 mSv.[9] A typical chest x-ray exposes a person to 100 μSv.[10] How many chest x-rays’ worth of radiation were the workers exposed to?
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28. 0.4 ms, 0.04 ms, 0.004 ms; 400 μs, 40 μs, 4 μs the ratio of the wavelengths of red and infrared is 7 to 100;
29. the ratio of the wavelengths of infrared and red is around 14 to 1
30. this is equivalent to 2,500 chest x-rays
- 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↵
- https://www.kgw.com/article/news/local/oregon-company-get-approval-to-build-nuclear-power-plants/283-7b26b8cd-12d5-4116-928a-065731f7a0f6 ↵
- https://en.Wikipedia.org/wiki/Nuclear_weapon_yield ↵
- https://www.noction.com/blog/network-latency-effect-on-application-performance ↵
- http://labman.phys.utk.edu/phys222core/modules/m6/The%20EM%20spectrum.html ↵
- https://en.Wikipedia.org/wiki/Chernobyl_disaster ↵
- https://www.cancer.org/treatment/understanding-your-diagnosis/tests/understanding-radiation-risk-from-imaging-tests.html ↵