Exponents: Scientific Notation
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What is scientific notation?
Scientific notation is a method for converting unwieldy numbers, whether huge or tiny, into a more manageable format. This conversion is done by using exponents; the format of the converted value will be a.bcdef...×10n, where a, b, c, d, e, f, etc, are numerals, and n is a positive or negative integer.
Why does scientific notation exist?
Scientists are often working with very large and very small numbers. Rather than writing out the entire number, scientific notation allows for shortening the expressions. For instance, instead of saying that a water molecule is about 0.00000000000275 meters across, we can say that its width is about 2.75×10−12 meters.
In day-to-day life, most people don't encounter gigantic or tiny numbers. Scientific notation was invented to make life easier for scientists, who often deal with astronomical numbers (in the literal sense of measuring distances between stars) and microscopic numbers (in the literal sense of needing some sort of microscope to see what they're measuring). Scientific notation makes things easier to handle.
How does scientific notation work?
The format for writing a number in scientific notation is fairly simple: (first [non-zero] digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10, raised to an appropriate power).
To convert a big number to scientific notation, follow these steps:
- Take the original value, and move the decimal point from after the last digit to after the first digit. For instance, take 7,375,498,555 and move the (understood) decimal point from after the final 5 to after the 7, to get 7.375498555.
- Count the number of spots that the decimal point has been moved. In this example, the point moved 9 places.
- Multiply the converted number by 10, raised to the power of however many places the point has been moved. In this example, the scientific notation is 7.375498555×109.
The process for converting a small number to scientific notation is similar:
- Take the original value, and move the decimal point from its original location to the spot right after the first non-zero digit. For example, take 0.00000000000275 and move the decimal point from after the first zero to right after the 2, to get 2.75.
- Count the number of spots that the decimal point has been moved. In this example, the point moved 12 places.
- Multiply the converted number by 10, raised to the power of the negative of however many places the point has been moved. In this example, the scientific notation is 2.75×10−12.
Whether you're converting large or small numbers to scientific notation, the part that comes before the "times 10 to some power" is called the coefficient; the 10 is the base; and the power is, well, the power.
How do you know what sign to use for the power on 10?
A large number is going to involve 10 to some positive value, such as 107, which is 1,000,000. A small number is going to involve 10 to some negative value, such as 10−7, which is 0.0000001. So expect positive powers when you're converting big numbers to scientific notation, and negative powers when you're converting small numbers.
In practice, the conversion process is fairly simple.
- Write 124 in scientific notation.
This is not a very large number, but it will work nicely for an example of how to convert to scientific notation.
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 I will have an equal value once I tack on the base and the power.
To convert 1.24 back to 124, I would multiply by 100: (1.24)(100) = 124. And 100 = 102.
Then, in scientific notation, 124 is written as:
1.24 × 102
In the example above, I used mathematical reasoning to explain why the power on 10 needed to be 2. But 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.
- Write in decimal notation: 3.6 × 1012
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:
So my answer is:
3,600,000,000,000
...or 3.6 trillion.
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".
- Write 0.000 000 000 043 6 in scientific notation.
(Note: The spaces between each triple of digits after the decimal point serve to make the number easier to read. They have no mathematical significance, any more than the commas in large numbers "mean" anythin. They're just a nicety.)
In scientific notation, the coefficient (that is, 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 given number is written as:
4.36 × 10−11
- Convert 4.2 × 10−7 to decimal notation.
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.
Then my answer is:
0.000 000 42
- Convert 0.000 000 005 78 to scientific notation.
This is a small number, so the exponent on 10 will be negative. The first "interesting" (that is, non-zero) 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 my answer is:
5.78 × 10−9
- Convert 93,000,000 to scientific notation.
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 my answer is:
9.3 × 107
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 × 104
−36,000 = −3.6 × 104
Don't confuse these!
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":
- Simplify and express in scientific notation: (2.6 × 105) (9.2 × 10−13)
Since I'm multiplying, I can move things around and simplify some of this stuff easily:
(2.6 × 105) (9.2 × 10−13)
= (2.6) (105) (9.2) (10−13)
= (2.6) (9.2) (105) (10−13)
= (2.6) (9.2) (105−13)
= (2.6) (9.2) (10−8)
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 × 101
Putting it all together, I have:
(2.6 × 105) (9.2 × 10−13)
= (2.6) (9.2) (10−8)
= (2.392 × 101) (10−8)
= (2.392)(101) (10−8)
= (2.392) (101−8)
= 2.392 × 10−7
Then (2.6 × 105) (9.2 × 10−13) = 2.392 × 10−7
Dividing numbers in scientific notation works about the same way.
- Simplify and express in scientific notation: (1.247 × 10−3) ÷ (2.9 × 10−2)
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 × 102)
= (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
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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