Exponents: Engineering Notation
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"Engineering" notation is very similar to scientific notation, except that the power on ten can only be a multiple of three. In this way, numbers are always stated in terms of thousands, millions, billions, etc. For instance, 13,460,972 is thirteen million and some. In the newspaper, it would probably be abbreviated as "13.5 million". In engineering notation, you would move the decimal point six places to the left to get 13.460972 × 106. Once you get used to this notation, you recognize that 106 means "millions", so you would see right away that this is around 13.5 million.
Every time you see a reference to some number of millions or billions or trillions, rather than a complete enumeration of the entire number with all its digits, the writer is, in effect, using engineering notation.
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Express 472,690,128,340 in engineering notation.
This is a twelve-digit number. I need to move the decimal point from the end of the number toward the beginning of the number, but I must move it in steps of three decimal places.
In this case, I must move the decimal point to between the 2 and the 6 (that is, to the location in the original number of the first comma), because this will leave nine digits (and nine is a multiple of 3) after the decimal point, and no more than three digits before the decimal point.
(Yes, 12 is a multiple of 3, but if I move the decimal point twelve places to the left, I'll have no non-zero digits to the left of the decimal point. This would be wrong. I have to have non-zero stuff to the left of the dot so, in this case, I have to stop at nine decimal places.)
This is a large number and I moved the decimal point nine places, so the power on 10 will be a positive 9. Then the answer is:
472.690128340 × 109, or 472.7 billion.
It may be simpler to think in terms of commas when converting larger numbers to engineering notation, because those commas are placed specifically in "regular" numbers in order to demarcate hundreds from thousands, thousands from millions, millions from billions, and so forth. So just look for the comma furthest to the left, and move the decimal point to that location.
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Express 83,201 in engineering notation.
I need to move the decimal point over to the left in sets of three digits (that is, in comma-delimited groups of digits). I can't move the decimal point any further than to the left of the 2, which is three places, so the answer is:
83.201 × 103, or 83.201 thousand.
When working with small numbers (that is, with numbers whose interesting digits are all the right of the decimal point), we don't have commas, but we can still think in terms of sets of three.
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Express 0.000 063 8 in engineering notation.
I need to move the decimal point over in sets of three. If I move the decimal point to the right three places, I'll be left with "0.0638", which won't do, because it'll leave me with just zero to the left of the decimal point. If I move the decimal point to the right nine places, I'll get "63800", which is too many digits to the left of the decimal point. So I need to move the decimal point six places.
Since this started out as a small number, the power on 10 will be negative; since I moved the decimal point six places, the power will be a negative 6. Then the answer is:
63.8 × 10–6, or 63.8 millionths.
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Express 0.397 53 in engineering notation.
I need to move the decimal point to the right three places. Since this started as a small number, the power on 10 will be negative:
397.53 × 10–3, or 397.53 thousandths.
You should notice that, in engineering notation, it is perfectly okay to have more than one digit to the left of the decimal point; in fact, you should expect to have something other than always only one digit. Just make sure that the power on 10 is a multiple of three.
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