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# Scientific Notation

last edited by 13 years, 6 months ago

# 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 × 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?

• 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

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.

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

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  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

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

• 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 10- 3.1 x 103 5.98 x 104

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

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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