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.)  

 

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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.

 

 

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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 10⁵

 

 

 

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

 

 

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How do you convert scientific notation back into standard notation? 

 

• Changing a number in scientific notation back into standard notation is pretty easy. It's just the reverse of what you have just done.

 

Let's change 5.63 x 10⁴ 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 10⁴  = 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.

 

 

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Examples of Scientific Notation

 

 

• 652,000            =       6.52 x 10⁵
   
   
• 2,560,000         =       2.56 x 10⁶
   
   
• 0.000001          =       1 x 10^-6
   
   
• 3.00568            =       3.00568 x 10⁰
   
   
• 5,529.5             =       5.5295 x 10³
   
   
• 159,000,630    =       1.5900063 x 10⁸
   
   

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/mr-scnot.html

 

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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 10²³ , we can just write that down.
   
   

 100,000,000,000,000,000,000,000 = 1 x 10²³

 

 

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

 

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Adding and Subtracting with Scientific Notation

 

• Not only can you write in scientific notation, you can solve equations with it too. Let's start with adding.

 

Here is an example of an equation: 2.76 x 10⁴ + 6.3 x 10².

 

First, we must change our exponents of ten so that they are both the same. We can change 10² into 10⁴ by the decimal of 6.3 to the left two times.

 

 

6.3 x 10² = .063 x 10⁴

 

 

Now our equation says: 2.76 x 10⁴ + .063 x 10⁴. 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 10⁴ + 6.3 x 10² = 2.823 x 10⁴

 

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• Subtracting in scientific notation is basically the same as addition.

 

Let's subtract 6.2 x 10⁴ and 3.1 x 10³. Again, we have to make the exponents the same. We can change 10³ to 10⁴ by moving the decimal in 3.1 to the left once.

 

 

3.1 x 10³ = 0.31 x 10⁴

 

 

Now our equation says 6.2 x 10⁴ - 0.31 x 10⁴. 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 10⁴ - 3.1 x 10³ = 5.98 x 10⁴

 

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Multiplying and Dividing with Scientific Notation

 

• You can also multiply and divide with scientific notation.
   

Let's multiply (2.12 x 10³)(3.5 x 10²). Because we are dealing with all multiplication, we can use the  Associative Property of Multiplication.

 

 

(2.12 x 10³)(3.5 x 10²) = (2.12 x 3.5)(10³ x 10²)

 

 

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 10⁵, and since 1 < 7.42 < 10, we can use it in our answer.

 

 

(2.12 x 10³)(3.5 x 10²) = 7.42 x 10⁵

 

*******************************************************************************************

 

• Because you can multiply in scientific notation, you can divide with it too.
   
   

Let's divide 6.3 x 10³ and 2.1 x 10². First, we can rewrite the problem.

 

 

 6.3 x 10³           6.3       10³

_________  = ____ x ____

 2.1 x 10²           2.1       10²

 

 

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 10¹, and since 1 < 3 < 10, we can use it in our answer.

 

 

 6.3 x 10³           

_________ = 3 x 10¹

 2.1 x 10²     

 

 

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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.

 

• http://www.fordhamprep.com/gcurran/sho/sho/lessons/lesson25.htm
   
   
   
• http://www.purplemath.com/modules/exponent3.htm
   
   
   
• http://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html
   
   
   
• http://www.astro.washington.edu/labs/clearinghouse/labs/Scimeth/mr-scnot.html (This link is also posted above, in "Examples of Scientific Notation.") 
   
   
   
• http://www.ieer.org/clssroom/scinote.html