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Chapter 5—Powers, Exponents, and Roots
5-8 Powers-0f-10 and Engineering Notataion
Engineering notation is a special case of power-of-10 notation. Even if you have no plans for becoming a scientist or engineer, you must master the procedures and terminology for engineering notation. Why? One very good reason is that it is that very foundation of the metric system of measurement — the system of measurement used throughout most of the world, with the notable exception of the United States. And even in the U.S., the metric system and engineering notation are used in most technical areas, including medicine, information technology, electronics, building trades, and growing numbers of other occupations.
Introduction to Engineering Notation
| Definition  Engineering notation is a special form of power-of-10 notation where - The coefficient must be between one and three digits (not including zero) to the left of the decimal point.
- The exponent must be 0 or any multiple of 3
And since this is a power-of-10 system of notation, the base is always 10. |
Examples
Are the the following values expressed in proper (normalized) engineering notation? Explain your response.
| 2.4 x 103 | Yes, because there is one non-zero digit to the left of the decimal point in the coefficient, and the notation allows between one and three digits in that location.. |
| 100.48 x 106 | because there are three non-zero digit to the left of the decimal point in the coefficient, and the notation allows between one and three digits in that location. |
| – 4.5 x 106 | Yes, because there is one non-zero digit to the left of the decimal point in the coefficient, and the notation allows between one and three digits in that location.. |
| 0.25 x 103 | No, because the digit to the left of the decimal point in the coefficient is a zero. |
| 1024 x 10-6 | No, because there are too many digits in the coefficient. |
| Note Regarding digits in the coefficient of normalized engineering notation: The number of digits to the right of the decimal point isn't important. The number of digits to the left of the decimal point must be between one and three digits. This means the actual value of those digits must be between 1 and 999. 
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Examples
Are the the following values expressed in proper (normalized) engineering notation? Explain your response.
| 2.4 x 102 | No, because the exponent must be zero or a multiple of three. |
| 100.48 x 106 | Yes. The coefficient is correct and the exponent is a multiple of three. |
| – 4.5 x 10-12 | Yes. The coefficient is correct and the exponent is a multiple of three. |
| 0.25 x 10-8 | No, because the exponent must be zero or a multiple of three. |
| 124 x 10-6 | Yes. The coefficient is correct and the exponent is a multiple of three. |
| Note Regarding the values of the exponent: 
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Examples & Exercises
Identifying Examples of Normalized
Engineering Notation Use these interactive Examples & Exercises to strengthen your understanding and build your skills: | |
What is so Special About Engineering Notation?
The engineering system of power-of-10 notation is the foundation for the world's most popular system of measurement -- the metric system. The Notation column in the table below shows the common range of powers of ten for the metric system. Notice that the exponents are 0 or multiples of 3. The column for Decimal Value shows the actual value of the corresponding power of ten. The remaining items in the table are important for working specifically with the metric system, but have no actual meaning for basic engineering notation.
| Notation | Prefix | Symbol | Scale | Decimal Value |
| 1012 | tera– | T | Trillion | 1 000 000 000 000 |
| 109 | giga– | G | Billion | 1 000 000 000 |
| 106 | mega– | M | Million | 1 000 000 |
| 103 | kilo– | k | Thousand | 1 000 |
| 100 | (none) | (none) | One | 1 |
| 10–3 | milli– | m | Thousandth | 0.001 |
| 10–6 | micro– | m | Millionth | 0.000 001 |
| 10–9 | nano– | n | Billionth | 0.000 000 001 |
| 10–12 | pico– | p | Trillionth | 0.000 000 000 001 |
You are probably familiar with a few of the more common units of metric measurement. How abot:
2 meters, 1 kilometer, 30 grams, 550 milligrams....?
You will learn a lot more about the specifics of the metric system in a later lessons. You will be better prepared for those lessons if you now make the effort to master the details of engineering notation -- the notation that is the foundation for the metric system of measurement.
Rewriting Decimals and Power-of-10 Values in Engineering Notation
Any number can be rewritten in engineering notation by moving a decimal point to the proper location.
| IMPORTANT Placing and moving a decimal point in a coefficient is actually a matter of multiplying or dividing the value by a factor of 10. Moving the decimal point one place to the left is actually dividing the value by a factor of ten. Moving the decimal point one place to the right is actually multiplying the value by a factor of 10. |
4308.2 is the same as 43082 ¸ 10
430.82 is the same as 43082 ¸ 100
43.082 is the same as 43082 ¸ 1000
4.3082 is the same as 43082 ¸ 10000
A common procedure in career mathematics is to adjust decimal values to conform to the standards of engineering notation.
Example
Consider this value expressed in a simple power-of-10 notation: 23.45 x 104.
- First, look at the exponent, 4. It should be changed to something such as 3 or 6. Let's try 3.
- Recall that when you decrease an exponent, you must move the decimal point to the right. So in this example: 23.45 x 104 becomes 234.5 x 103
- ... and 234.5 x 103 is a valid expression in engineering notation — there are between one and three digits on the left side of the decimal point in the coefficient, and the exponent is a multiple of three.
But what about changing the exponent in 23.45 x 104 from 4 to 6, instead of from 4 to 3. Watch what happens:
Increasing the exponent by 2 means you must move the decimal point in the coefficient two places to the left. So 23.45 x 104 becomes .2345 x 106.
... but this is not really a valid expression of engineering notation. The exponent is, indeed, a multiple of 3, but there are no digits (remember, 0 doesn't count) on the left side of the decimal point in the coefficient.
- 23.45 x 104 is not expressed in proper engineering notation
- 0.2345 x 106 does not express the the value in proper engineering notation.
- 234.5 x 103 is a proper expression of scientific notation.
More Examples
| Remember - When you move the decimal point in the coefficient to the right, you must also decrease the power-of-10 exponent by the same number of units.
- When you move the decimal point in the coefficient to the left, you must also increase the power-of-10 exponent by the same number of units.
- When you increase the value of the power-of-10 exponent, you must also move the decimal point the same number of units to the left.
- When you decrease the value of the power-of-10 exponent, you must also move the decimal point the same number of units to the right.
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Examples & Exercises
Rewriting Values with Engineering Notation Use these interactive Examples & Exercises to strengthen your understanding and build your skills: | |
