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1.3: Scientific and Engineering Notation

  • Page ID
    25016
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    Scientific and engineering notations are ways to express numbers without a lot of trailing or leading zeroes. They also simplify calculations. Before going any further, it would be a good idea to obtain a proper scientific calculator, such as the ones shown in Figure 1.3.1 . The simplest can be had for $10 to $20, while more powerful graphing types will run ten times as much (although good buys can often be found on the used market). Some of the brands to consider are Casio, Sharp and Texas Instruments. If you plan to continue study into AC electrical circuits, it would be wise to make sure the calculator can solve simultaneous equations with complex coefficients.

    clipboard_e5572edddecf582f1bf86f07b984e5539.png

    Figure 1.3.1 : Scientific calculators.

     

    The idea behind scientific notation is to represent the value in two parts: a precision portion, or mantissa; and the magnitude, a power of ten called the exponent. The representation looks like this (\(m\) is the mantissa and \(e\) is the exponent)

    \(m \times 10^n\)

    • Thus, 360, which is \(3.6 \times 100\), can be written as \(3.6 \times 10^2\), where 3.6 is the mantissa and 2 is the exponent
    • Similarly, 0.00275, which is \(2.75 \times 0.001\), can be written as the value \(2.75 \times 10^{−3}\).

    As the base of the exponent is always 10, a more compact form replaces the “\(\times 10\)” with “E”. Thus, these two values can be written as 3.6E2 and 2.75E−3. When adding or subtracting values in this form, the first step is to make sure that all of the values have the same exponent. Then, the precision portions are simply added together. For example:

    • 3.6E2 + 1.1E2 is equal to 4.7E2.
    • Similarly, 3.6E2 + 5E1 is converted as 3.6E2 + 0.5E2 yielding 4.1E2. Alternately, it can be converted as 36E1 + 5E1 yielding 41E1 (the same answer, or 410 in ordinary form).

    Where this notation is particularly useful is when multiplying or dividing. For multiplying, multiply the mantissas and add the exponents. For dividing, divide the mantissas and subtract the exponents. For example, multiply 20000 by 360000. This is equivalent to 2E4 times 3.6E5. The result is 7.2E9 (i.e., 7200000000). Similarly, dividing the value 0.006 by 50000 yields 6E−3 divided by 5E4, or 1.2E−7 (or in ordinary form 0.00000012). Notice the cumbersome and error-prone quantities of trailing and leading zeroes in these examples when using ordinary form. With scientific notation, we no longer have to deal with them.

    Example 1.3.1

    Convert the following numbers into normalized scientific notation

    1. 2100
    2. 0.005
    3. 32,000,000
    4. 0.0000741
    Answer a

    \(2.1 \times 10^3\)

    Answer b

    \(5 \times 10^-3\)

    Answer c

    \(3.2 \times 10^7\)

    Answer d

    \(7.4 \times 10^-5\)

     

    Engineering notation is similar to scientific notation with the caveat that the exponent must by a multiple of 3. Thus, 390000 would be written as either 390E3 or possibly as 0.39E6. Each multiple of 3 has a name and abbreviating letter to make the written representation even more compact. You are probably already familiar with many of them. For example, \(10^3\) or 1000, is kilo which is abbreviated as \(k\). Also, \(10^6\), or one million, is Mega, and abbreviated \(M\) (note capital letter).

    Table 1.3.1 : Common engineering notation prefixes and abbreviations.
    Exponent Name Abbreviation
    12 Tera T
    9 Giga G
    6 Mega M
    3 kilo k
    -3 milli m (note lower case)
    -6 micro \(\mu\) (Greek letter \(mu\))
    -9 nano n
    -12 pico p

    A partial list of exponents, their names, and abbreviations is shown in Table 1.3.1 . There are standards for both larger and smaller exponents, but this table is something that should be committed to memory as they are in wide use.

    Example 1.3.1

    Convert the following into engineering notation using appropriate prefixes.

    1. 2100 grams
    2. 0.005 meters
    3. 32,000,000 bits per second (bps)
    4. 0.0000741 seconds
    Answer a

    2.1 kilograms (2.1 kg)

    Answer b

    5 millimeters (5 mm)

    Answer c

    32 Megabits per second (32 Mbps)

    Answer d

    74.1 microseconds (74.1 \(\mu\)s)

    Notice that the final answers are much more compact and less error prone.

    After some initial effort, certain common shortcuts will become second nature. For example (and keeping it generic), micro times kilo will yield milli (i.e., millionths times thousands yields thousandths). Similarly, the reciprocal of kilo is milli and the reciprocal of micro is Mega. Further, milli divided by kilo yields micro, Mega divided by kilo yields kilo, and so on. Gaining facility with these kinds of shortcuts will allow you to make estimates in your head very quickly, a much desired skill.


    This page titled 1.3: Scientific and Engineering Notation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James M. Fiore via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.