Scientific Notation
What is scientific notation?
 Scientific notation is a special form of writing numbers. (Usually used for either really big numbers or really small numbers.)
What does scientific notation look like?
 Scientific Notation is based on powers of ten. The basic notation for scientific notation is shown in this algebraic expression:
a × 10^{b}
"a", in the expression is the coefficient. It is always written in decimal form. The rule for "a" is that it must be 1 or more and also must be under 10.
1 < a < 10
"b", in the expression is the exponent for 10, which is the base. "b" must be an integer, either positive or negative.
How do you write in scientific notation?
 To first figure out the the coefficient ("a"), you first write down the first digits of the number you are converting that come before multiple zeros (or just one zero).
If the number you are converting doesn't have any zeros at the end, then you write the entire number down. You then place a decimal point right after the first number.
example: 256,000 → 256,000 2.56
↓ ↓
first digits of number decimal point after first digit
That is for the first part of the expression, but how do you write the second?

As you have seen, the second part of the expression is the base (10) with an exponent. Since you already know the base, how do you find the exponent?
Let's compare the decimal number and the original number:
2.56 and 256,000.
You can see that there are 5 decimal places between each number. That means the exponent of ten is 5. Also, because the decimal moves to the right, the number is positive.
2(.)56000.
>>>>> 5 decimal places to the right
So this is how you write 256,000 is scientific notation:
256,000 = 2.56 x 10^{5}
The example we just saw was for a really big number, but what about a small number?
Let's use 0.00054 as an example.
For smaller numbers, you take the first number that isn't a zero and everything after it. Then you put a decimal point after the first number.
0.00054 → 54 5.4
↓ ↓
number that isn't a zero and everything after decimal point after first digit
For the second half, we again compare the new number and the original number.
5.4 and 0.00054
This time we can see that 4 decimal places going to the left.
0.0005(.)4
<<<< 4 decimal places to the left
Because the decimal moves to the left, instead of the exponent being 4, it's negative four. Our final answer is this:
0.00054 = 5.4 x 10^{4}
How do you convert scientific notation back into standard notation?
Let's change 5.63 x 10^{4 }back into standard form. We know that the exponent above 10 is the number of places the decimal moves. Because 4 is a positive number, we move the decimal to the right 4 times.
5(.)6300.
>>>> 4 decimal places to the right.
We end up with 56,300 , which is our final answer.
5.63 x 10^{4 } = 56,300
Changing a number in scientific notation that has a negative exponent over the ten is pretty much the same. The only difference is that the decimal moves to the left instead of the right.
Examples of Scientific Notation

652,000 = 6.52 x 10^{5}

2,560,000 = 2.56 x 10^{6}

0.000001 = 1 x 10^{6}

3.00568 = 3.00568 x 10^{0}

5,529.5 = 5.5295 x 10^{3}

159,000,630 = 1.5900063 x 10^{8}
If you want a chart of scientific notation examples with numbers used in real life, you can find one at http://www.astro.washington.edu/labs/clearinghouse/labs/Scimeth/mrscnot.html
Why use scientific notation?
Why in the world is scientific notation used? Is it just a fancy way to write numbers differently?
Actually, scientific notation is a method of writing either really big numbers or really small numbers. It is favored by both mathematicians and scientists.

For example, astronomers usually have to deal with really big numbers, such as 100,000,000,000,000,000,000,000 (One hundred trillion billion), which is the approximate number of stars in the universe. That's a really big number! It would be tiring and boring to write it over and over again. This is the reason we like to use scientific notation. Because we know that 100,000,000,000,000,000,000,000 is equal to
1 x 10^{23 }, we can just write that down.
100,000,000,000,000,000,000,000 = 1 x 10^{23}
We have just seen an example of a large number, but remember, scientific notation is also used in really small numbers too.
 For example, the diameter of an atom is approximately 0.00000000000001 km. Instead of writing this over and over again, scientists would rather write in scientific notation.
0.00000000000001 = 1 x 10^{14}
Number of Stars in the Universe Diameter of an atom
http://mrdowling.com/601maps.html http://gpc.edu/~pgore/PhysicalScience/Atoms.html
Adding and Subtracting with Scientific Notation
Here is an example of an equation: 2.76 x 10^{4 }+ 6.3 x 10^{2}.
First, we must change our exponents of ten so that they are both the same. We can change 10^{2 }into 10^{4 }by the decimal of 6.3 to the left two times.
6.3 x 10^{2 }= .063 x 10^{4}
Now our equation says: 2.76 x 10^{4 }+ .063 x 10^{4}. To add this, we can leave the base and exponents the same, and add the coefficients.
2.76^{ }+ .063 = 2.823
If our answer didn't come out to 1 or more and less than 10, we would have to change the decimal and exponent of 10. But because 1 < 2.823 < 10, we can use it in our final answer.
2.76 x 10^{4 }+ 6.3 x 10^{2 }= 2.823 x 10^{4}
********************************************************************************************
Let's subtract 6.2 x 10^{4 }and 3.1 x 10^{3}. Again, we have to make the exponents the same. We can change 10^{3} to 10^{4 }by moving the decimal in 3.1 to the left once.
3.1 x 10^{3 }= 0.31 x 10^{4}
Now our equation says 6.2 x 10^{4 }^{ }0.31 x 10^{4}. We then subtract the coefficients:
6.2  0.31 = 5.89
Because 1 < 5.89 < 10, we can use it in our answer.
6.2 x 10^{4 } 3.1 x 10^{3 }= 5.98 x 10^{4}
Multiplying and Dividing with Scientific Notation
Let's multiply (2.12 x 10^{3})(3.5 x 10^{2}). Because we are dealing with all multiplication, we can use the Associative Property of Multiplication.
(2.12 x 10^{3})(3.5 x 10^{2}) = (2.12 x 3.5)(10^{3 }x 10^{2})
We then simplify the expression. (You can leave 10 as the base and add the exponents for the second half of the expression.)
We then get 7.42 x 10^{5}, and since 1 < 7.42 < 10, we can use it in our answer.
(2.12 x 10^{3})(3.5 x 10^{2}) = 7.42 x 10^{5}
*******************************************************************************************
Let's divide 6.3 x 10^{3 }and 2.1 x 10^{2}. First, we can rewrite the problem.
6.3 x 10^{3 }6.3 10^{3}
_________ = ____ x ____
2.1 x 10^{2 }2.1 10^{2}
We then simplify the expression. (You can leave 10 as the base and just subtract the exponents when dividing the second half of the expression.)
We then get 3 x 10^{1}, and since 1 < 3 < 10, we can use it in our answer.
6.3 x 10^{3 }
_________ = 3 x 10^{1}
2.1 x 10^{2 }
References
These are a few of the sites I got my information on that I used the most. If you had trouble understanding some parts of my page, you should visit these web sites for more help.
Comments (16)
Anonymous said
at 3:14 pm on Oct 24, 2007
You spelled "algebraic" wrong.
Anonymous said
at 3:53 pm on Oct 24, 2007
...oh...thanks.
Anonymous said
at 9:53 pm on Oct 29, 2007
Scientific notation i thought we were in math... whatever
Anonymous said
at 5:25 pm on Oct 31, 2007
lol nick. Nice page.... um...its long.. um... theres words on it... lol. Seriously, good job. Comment on my page plz :)
Anonymous said
at 5:27 pm on Oct 31, 2007
o ya now i know what i was gnna say. The words are a little small, so its kinda harder to read. If the words were a little bigger, ur page would be amazing in my opinion
Anonymous said
at 1:14 pm on Nov 5, 2007
Nice pictures, also i like how you divided different parts of the scientific notation
Anonymous said
at 1:17 pm on Nov 5, 2007
nice info, try 2 lighten it up a bit! ^O^
Anonymous said
at 1:31 pm on Nov 5, 2007
great job with all the examples.
Anonymous said
at 2:27 pm on Nov 5, 2007
good job its perfect
Anonymous said
at 2:27 pm on Nov 5, 2007
Great Page you are a sure to get an A++ . . . +
Anonymous said
at 2:53 pm on Nov 5, 2007
woooooooooooooooooooooooooooooooooooooooooooooooooooo that's a Nice page! Just seems like you are saying the same thing over and over.
Anonymous said
at 5:27 pm on Nov 5, 2007
u write long comments... i just noticed that ;]
Anonymous said
at 10:15 pm on Nov 5, 2007
i agree with mike, and it kinda rambles on. Its just my opinion, but it's hard to read all that information. Probably just cuz I am not good at comprehending a lot of information though. It's an awesome page! Definite A!
Anonymous said
at 10:43 pm on Nov 5, 2007
Some good information. I notice that you exchange a and A and b and B. Mathematicaly these are different symbols, representing different values.
Anonymous said
at 10:46 pm on Nov 5, 2007
Standard form is different from Scientific Notation. 1,258 is standard form for 1.258 x 10^3.
Anonymous said
at 10:39 pm on Nov 14, 2007
Fabulous page! Excellent information clearly presented.
One little grammar error or typo near the begining. ""b" must be an integer, either positive of negative"
4 of 4 points.
12 of 12 project points.
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