Scientific Notation


 

Scientific Notation 

 

 

What is scientific notation?

 

 

 

 

What does scientific notation look like?

 

 

 

 a × 10b

 

 

 

"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?

 

 

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?

 

 

 

 

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

 

 

 

 

How do you convert scientific notation back into standard notation? 

 

 

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.

 

 

 

 

Examples of Scientific Notation

 

 

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

 

 

 

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.

 

 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.

 

 

 

0.00000000000001 = 1 x 10-14

 

 

    Number of Stars in the Universe                             Diameter of an atom

 

 Image Preview                    image

 

 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 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

 

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

 

 

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

 

 

 

Multiplying and Dividing 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

 

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

 

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     

 

 

 

 

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.