Powers of 10, Scientific/Engineering Notation, and Metric Prefixes

This entry is meant to help you understand powers of 10, scientific notation, engineering notation, and some metric prefixes.

Powers of 10, Scientific Notation vs. Engineering Notation

We have all probably seen something written in math class or in reading any kind of electronics text that looks like this: \(4.32 \times 10^5\) This is called scientific notation, or a power of 10. In this case, it would be 4.32 times 10 to the power of 5. The number 10 is called the base, the number 5 is called the exponent. The exponent tells us how many times the base is to be multiplied by itself. In our example, \(\times 10^5\) means 10x10x10x10x10.
In Scientific Notation the exponent can be any number, even negative numbers and zero. We will talk about how Engineering Notation differs from Scientific Notation in just a moment, but first, let's find out out exactly what \(4.32 \times 10^5\) means.
First, we see that our base is 10 and our exponent is 5, so we know that we need to multiply the base (10) by itself equal to our exponent (5), so like mentioned before 10x10x10x10x10. That makes our expression 4.32 x 10 x 10 x 10 x 10 x 10 = 432,000. Using this formula, we are able to express very large numbers and very small numbers much more efficiently than we would otherwise be able to. Consider an electric charge of 1 Coulomb, which has a proton or electron count (depending on polarity) of roughly \(6.25 \times 10^{18}\). It is much more efficient, and less prone to error in calculations if we express this number in scientific notation as opposed to writing it out fully.

\(6.25 \times 10^{18}\)
6,250,000,000,000,000,000

Both of these numbers are equal, just that one is expressed in Scientific Notation.

Notation With Negative Exponents

So far, we have only talked about a number where the exponent is positive, but when we express numbers that are smaller than 1.0, like 0.025, we have to use a negative exponent. In our example of 0.025, we would say \(2.5 \times 10^{-2}\). This is the equivalent of saying \(2.5 \div 10 \div 10\). This gives us a quotient of 0.025!

So, to recap, positive exponent means multiply the base (10) by itself the number of times indicated by the exponent. A negative exponent means divide the base (10) by itself the number of times indicated by the exponent.

But what, you might ask yourself, happens when you have a very large or very small number? Am I to sit here with a number like \(5.81 \times 10^{21}\) and multiply 10 by itself 21 times? HOW TEDIOUS! Well no, there is an easier way my friends, and it works with both positive and negative exponents. Let's consider our example of \(5.81 \times 10^{21}\). The exponent is positive, so our number is going to get larger. If the exponent was negative, we would know that our number is going to get smaller (because we're dividing). What happens to a decimal place when a number gets larger? It moves to the right. If we wanted to multiply 5.81 times 10, we would get 58.1 right? Well, that single move to the right could be expressed as \(5.81 \times 10^1\) because we multiplied the base (10) by itself 1 time, or 5.81 x 10 = 58.1. If we had said \(5.81 \times 10^2\) that would be the same as saying 5.81 x 10 x 10 = 581.0 right? Our decimal place is moving to the right equal to the exponent, and as we progress beyond the last digit of our original number, we're adding zeros to fill in the holes.

If I take \(5.81 \times 10^{21}\) it's the same as saying "move the decimal place twenty-one times to the right". Since we only have 2 digits to the right of the decimal place, we'll have to add zeros as we go. We end up with 5,810,000,000,000,000,000,000. If our exponent had been negative, we would have moved the decimal place 21 times to the left and added zeros. \(5.81 \times 10^{-21}\) = 0.00000000000000000000581, which is a very small number but not uncommon in electronics.

Engineering Notation

Engineering Notation is essentially the same as scientific notation except that we only use exponents that are powers of 3. So exponents like -12, -9, -6, -3, 3, 6, 9, 12, and so on. The reason for this is that engineers use metric prefixes to express numbers. For example, we've all heard of a kilometer. Kilo is the metric prefix, and meter is the unit of measure. 1 kilometer is equal to 1,000 meters is equal to \(1.0 \times 10^3\) meters. In the engineering world, there are only metric prefixes for values that have an exponent which is a multiple of 3. A list of some of the metric prefixes can be found at:
http://tech99.net/wp-content/uploads/2011/06/SI-units.jpg
Typically speaking, a number expressed in Engineering Notation is always expressed as a number between 1 and 1,000 with a base of 10 and an exponent which is a multiple of 3. For example, \(5.81 \times 10^3\) units would be equal to 5.81 kilo-units or 5,810 units. Similarly, \(581.0 \times 10^{-3}\) units would be equal to 0.581 units which would be stated as "Five hundred and eighty-one milli-units". \(5.81 \times 10^{-6}\) would be 5.81 micro-units.

I hope you have a better understanding now of powers of 10, and how they might be used in electronics. Also, I hope that you understand why scientific notation/engineering notation helps us to display measurements more efficiently and in a way that makes calculations less prone to error. You will see powers of ten and metric prefixes very often in engineering.

For a discussion of arithmetic using scientific notation, see All About Circuits eBook, Volume I - DC, Arithmetic with Scientific Notation

References

Mitchel E. Schultz, Grob's Basic Electronics, McGraw-Hill, 2011, ISBN 978-0-07-351085-9

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