16 Other Conversions

You may use a calculator in this module as needed.

Converting Measurements of Time

Ominous message from Google Earth saying "Unfortunately, Earth has stopped."

You probably know all of the necessary conversions for time: 60 seconds in a minute, 24 hours in a day, etc.

When we get to units of time larger than weeks, however, we encounter problems because not all months have the same number of days, a year is not exactly 52 weeks, and the time it takes for the Earth to orbit the Sun is not exactly 365 days. Therefore, it doesn’t make sense to expect an exact answer to a question like “how many minutes are in one month?” We will have to use our best judgment in situations such as these.

1 min = 60 sec

1 hr = 60 min

1 day (dy) = 24 hr

1 week (wk) = 7 dy

1 year (yr) = 365 dy

Exercises

  1. How many minutes is one standard 365-day year?
  2. Have you been alive for one billion seconds? Is this even possible?

Converting Rates

Previously when we were converting units, we began with units in the numerator only. If we need to convert a rate, however, we’ll begin with units in both the numerator and denominator.

Exercises

Usain Bolt holds the world record time for the 100-meter dash, 9.58 seconds.

  1. What was his average speed in kilometers per hour?
  2. What was his average speed in miles per hour?

The more information we know, the more things we can figure out.

Exercises

An F-15 fighter jet can reach a sustained top speed of roughly Mach 2.3; this is 2.3 times the speed of sound, which is 770 miles per hour.[1]

  1. What is the jet’s top speed in miles per hour?
  2. At this speed, how many miles would the jet travel in one minute?
  3. The jet’s range at this speed before it runs out of fuel is around 600 miles. If the jet flies 600 miles at top speed, for how many minutes will it fly?
  4. The jet’s maximum fuel capacity is 3,475 gallons. If the jet flies 600 miles and burns 3,475 gallons of fuel, find the jet’s fuel efficiency, in miles per gallon.
  5. Rewrite the jet’s fuel efficiency, in gallons per mile.
  6. How many gallons of fuel does the jet consume in one minute?

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
10^{12} 10^{9} 10^{6} 10^{3} 10^{0}

Notice that the powers of these units are multiples of 3, just as with the engineering notation we saw at the end of Module 11. Each prefix is 1,000 times the next smaller prefix, so moving one place in the chart is equivalent to 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 2^{10} = 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.

Exercises

  1. A 5\frac{1}{4} inch floppy disk from the 1980s could store about 100 kB; a 3\frac{1}{2} inch floppy disk from the 1990s could store about 1.44 MB. By what factor was the storage capacity increased?
  2. How many times greater is the storage capacity of a 2 TB hard drive than a 500 GB hard drive?
  3. 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.”[2] How much electricity per home is this?
  4. 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?
  5. 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.[3] (Video of Tsar Bomba was declassified in 2020.) How many times more powerful was Tsar Bomba than Trinity?

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
10^{0} 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 \frac{1}{1000}. The negative exponents can sometime be complicated to work with, and it may help to think about things in reverse.

1  meter  = 10^{3} millimeters   = 10^{6} micrometers   = 10^{9} nanometers   = 10^{12} picometers

1 second = 10^{3} milliseconds = 10^{6} microseconds = 10^{9} nanoseconds = 10^{12} picoseconds

…and so on.

See https://physics.nist.gov/cuu/Units/prefixes.html for a list of more prefixes.

Exercises

  1. 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.”[4] Rewrite these times so that they are all in terms of milliseconds, then rewrite them in terms of microseconds.
  2. The wavelength of red light is around 700 nm. Infrared radiation has a wavelength of approximately 10 μm.[5] Find the ratio of these wavelengths.
  3. 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.[6] A typical chest x-ray exposes a person to 100 μSv.[7] How many chest x-rays’ worth of radiation were the workers exposed to?

a sign advertising a college's microelectronics program

Exercise Answers

  1. My sources for the following set of questions are a combination of former students in the Air National Guard and people who sound like they know what they're talking about on the internet, particularly in this Quora discussion. ↵
  2. Source: https://www.kgw.com/article/news/local/oregon-company-get-approval-to-build-nuclear-power-plants/283-7b26b8cd-12d5-4116-928a-065731f7a0f6 ↵
  3. Source: https://en.wikipedia.org/wiki/Nuclear_weapon_yield ↵
  4. Source: https://www.noction.com/blog/network-latency-effect-on-application-performance ↵
  5. Source: http://labman.phys.utk.edu/phys222core/modules/m6/The%20EM%20spectrum.html ↵
  6. Source: https://en.wikipedia.org/wiki/Chernobyl_disaster ↵
  7. Source: https://www.cancer.org/treatment/understanding-your-diagnosis/tests/understanding-radiation-risk-from-imaging-tests.html ↵

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Technical Mathematics, 2nd Edition Copyright © 2024 by Morgan Chase is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.