By using exponents, we can reformat numbers. This can be helpful, in much the same way that it's helpful (that is, it's easier) to write "twelve trillion" rather than 12,000,000,000,000., or "thirty nanometers" rather than "0.00000003 meters".
For very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers).
The format for writing a number in scientific notation is fairly simple: (first digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an appropriate power).
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The conversion is fairly simple.
This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first convert the "124" to "1.24". This is not the same number as what they gave me, but (1.24)(100) = 124 is, and 100 = 10^{2}.
Then, in scientific notation, 124 is written as 1.24 × 10^{2}.
Actually, converting between "regular" notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places. To do the conversion for the previous example, I'd count the number of decimal places I'd moved the decimal point. Since I'd moved it two places, then I'd be dealing with a power of 2 on 10. But should it be a positive or a negative power of 2? Since the original number (124) was bigger than the converted form (1.24), then the power should be positive.
Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need to move the decimal point to the right, in order to make the number LARGER. Since the exponent on 10 is "12", I'll need to move the decimal point twelve places over.
First, I'll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:
Then I fill in the loops with zeroes:
In other words, the number is 3,600,000,000,000, or 3.6 trillion
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Idiomatic note: "Trillion" means a thousand billion − that is, a thousand thousand million − in American parlance; the British-English term for the American "billion" would be "a milliard", so the American "trillion" (above) would be a British "thousand milliard".
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In scientific notation, the number part (as opposed to the ten-to-a-power part) will be "4.36". So I will count how many places the decimal point has to move to get from where it is now to where it needs to be:
Then the power on 10 has to be −11: "eleven", because that's how many places the decimal point needs to be moved, and "negative", because I'm dealing with a SMALL number.
So, in scientific notation, the number is written as 4.36 × 10^{−11}
Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a seven, I will be moving the decimal point seven places. Since I need to move the point to get a small number, I'll be moving it to the left.
The answer is 0.000 000 42
This is a small number, so the exponent on 10 will be negative. The first "interesting" digit in this number is the 5, so that's where the decimal point will need to go. To get from where it is to right after the 5, the decimal point will need to move nine places to the right. (Count 'em out, if you're not sure!)
Then the power on 10 will be a negative 9, and the answer is 5.78 × 10^{−9}
This is a large number, so the exponent on 10 will be positive. The first "interesting" digit in this number is the leading 9, so that's where the decimal point will need to go. To get from where it is to right after the 9, the decimal point will need to move seven places to the left.
Then the power on 10 will be a positive 7, and the answer is 9.3 × 10^{7}
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Remember: However many spaces you moved the decimal, that's the power on 10. If you have a small number in decimal form (smaller than 1, in absolute value), then the power is negative for the scientific notation; if it's a large number in decimal (bigger than 1, in absolute value), then the exponent is positive for the scientific notation.
Warning: A negative on an exponent and a negative on a number mean two very different things! For instance:
−0.00036 = −3.6 × 10^{−4}
0.00036 = 3.6 × 10^{−4}
36,000 = 3.6 × 10^{4}
−36,000 = −3.6 × 10^{4}
Don't confuse these!
You can use the Mathway widget below to practice converting a regular number into scientific notation. Try the entered exercise, or type in your own exercise. Then click the "paper-airplane" button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
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(Click here to be taken directly to the Mathway site, if you'd like to check out their software or get further info.)
You might be asked to multiply and divide numbers in scientific notation. I've never really seen the point of this, since, in "real life", you'd be dealing with these messy numbers by using a calculator, but here's the process, if you have to "show your work":
Since I'm multiplying, I can move things around and simplify some of this stuff easily:
(2.6 × 10^{5}) (9.2 × 10^{−13})
= (2.6) (10^{5}) (9.2) (10^{−13})
= (2.6) (9.2) (10^{5}) (10^{−13})
= (2.6) (9.2) (10^{5−13})
= (2.6) (9.2) (10^{−8})
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Okay; I've simplified the power-of-ten part. Now I have to deal with the 2.6 times 9.2, remembering to convert my product to scientific notation:
2.6 × 9.2 = 23.92 = 2.392 × 10 = 2.392 × 10^{1}
Putting it all together, I have:
(2.6 × 10^{5}) (9.2 × 10^{−13})
= (2.6) (9.2) (10^{−8})
= (2.392 × 10^{1}) (10^{−8})
= (2.392)(10^{1}) (10^{−8})
= (2.392) (10^{1−8})
= 2.392 × 10^{−7}
Then (2.6 × 10^{5}) (9.2 × 10^{−13}) = 2.392 × 10^{−7}
Dividing numbers in scientific notation works about the same way.
First, I'll deal with the powers-of-ten part:
(1.247 × 10^{−3}) ÷ (2.9 × 10^{−2})
= (1.247 ÷ 2.9) (10^{−3} ÷ 10^{−2})
= (1.247 ÷ 2.9) (10^{−3} × 10^{2})
= (1.247 ÷ 2.9) (10^{−1})
Now I'll deal with the division:
1.247 ÷ 2.9 = 0.43 = 4.3 × 10^{−1}
Putting it all together, I get:
(1.247 × 10^{−3}) ÷ (2.9 × 10^{−2})
= (1.247 ÷ 2.9) (10^{−1})
= (4.3 × 10−1) (10^{−1})
= (4.3) (10^{−1}) (10^{−1})
= (4.3) (10^{−2})
= 4.3 × 10^{−2}
So the answer is: (1.247 × 10^{−3}) ÷ (2.9 × 10^{−2}) = 4.3 × 10^{−2}
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If you are required to do problems like these, remember that you can always check your answers in your calculator. For instance, entering "1.247 EE −3 / 2.9 EE −2" on my calculator returns "0.043", which equals 4.3 × 10^{−2} in scientific notation. If you have to do a lot of these problems, you may find it useful to set your calculator to display all values in scientific notation. Check your owner's manual for instructions.
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