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Big Numbers and Small Numbers In Scientific Notation (exponential notation)
Chemists regularly use some pretty big numbers, some examples are given below:
 602000000000000000000000, is known as Avogadro's Constant
 101300 is atmospheric pressure in pascals (Pa)
 96484 is known as the Faraday Constant
 300000000 which is the speed of light in a vacuum in m s^{1}
Imagine trying to do calculations involving Avogadro's Constant, 602000000000000000000000, it is really hard to keep track of all those zeroes!
If you accidently leave off just one of those zeroes then the number you write will be 10 times smaller than it should have been.
If you lost two zeroes, then the number would be 100, times smaller than it should be!
If you lost three zeroes, then the number would be 1000 times smaller than it should be ....
and are you starting to see a pattern here?
602000000000000000000000 could be thought of as the product of a number multiplied by 10, or by 10 × 10 (100), or by 10 × 10 × 10 (1000), etc as shown below:
 602,000,000,000,000,000,000,000 = 60,200,000,000,000,000,000,000 × 10
 602,000,000,000,000,000,000,000 = 6,020,000,000,000,000,000,000 × 10 × 10
 602,000,000,000,000,000,000,000 = 602,000,000,000,000,000,000 × 10 × 10 × 10
 602,000,000,000,000,000,000,000 = 60,200,000,000,000,000,000 × 10 × 10 × 10 × 10
 602,000,000,000,000,000,000,000 = 6,200,000,000,000,000,00 × 10 × 10 × 10 × 10 × 10
Now if we had a shorthand way of representing all those "× 10s" we would have a useful way to keep track of all those zeroes.
Well mathematicians have developed this shorthand. They use a small superscript number to the right of the number 10 to tell us how many "× 10"s are needed to make up the number.
We are going to call this number the exponent.
The number "10" is just 1 "× 10" so the exponent is "1": 10^{1}, but usually we just write 10.
The number "100" is 1 "× 10 × 10", there are, "2" "×10s" so the exponent is "2": 10^{2}.
The number "1,000" is 1 "× 10 × 10 × 10", that is, "3" "×10s" so the exponent is "3": 10^{3}.
The number "10,000" is 1 "× 10 × 10 × 10 × 10", that is, "4" "×10s" so the exponent is "4": 10^{4}.
Do you see another pattern emerging?
We are replacing each "0" in the number with "×10" and the exponent increases by "1".
So we could just count the number of "0"s after the last nonzero digit in our original number to determine the value of the exponent.
So we could represent Avogadro's Constant, 602,000,000,000,000,000,000,000, as 602 with 21 "0"s after it as:
602 × 10^{21}
602 × 10^{21} is much easier to write than 602,000,000,000,000,000,000,000 but we can keep going as shown below:
602 × 10^{21} = 60.2 × 10 × 10^{21} = 60.2 × 10^{22}
60.2 × 10^{22} = 6.02 × 10 × 10^{22} = 6.02 × 10^{23}
We usually express Avogadro's Constant as 6.02 × 10^{23}.
When we represent numbers as small number between 0 and 10 (6.02) multiplied by 10 to the power of another number (× 10^{23}) we call this scientific notation or exponential notation.
The small number between 0 and 10 (6.02) is called the coefficient.
The "power" we raise the "10" to is called the exponent (23 in this case).
Chemists also work with some really small numbers. For example:
 the mass of an electron is 0.0000000000000000000000000000009109 kilograms
 the mass of a proton is 0.000000000000000000000000001673 kilograms
 an atom has a diameter of about 0.0000000002 metres
In these cases we divide the number by 10, let's take the diameter of an atom as an exmple:
 0.0000000002 = 0.000000002 ÷ 10
 0.0000000002 = 0.00000002 ÷ (10 × 10)
 0.0000000002 = 0.0000002 ÷ (10 × 10 × 10)
 0.0000000002 = 2 ÷ (10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10)
Which, using what we learnt above about exponents, we could also write as:
 0.0000000002 = 0.000000002 ÷ 10 = 0.000000002 ÷ 10^{1}
 0.0000000002 = 0.00000002 ÷ (10 × 10) = 0.00000002 ÷ 10^{2}
 0.0000000002 = 0.0000002 ÷ (10 × 10 × 10) = 0.0000002 ÷ 10^{3}
 0.0000000002 = 2 ÷ (10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10) = 2 ÷ 10^{10}
We don't usually express numbers as a division by 10 raised to some power, instead we write the number in scientific notation using multiplication of 10 raised to some power.
How can you convert a division to a muliplication?
Mathematicians have a trick for this too!
They use a minus sign () before the first number of the exponent.
Some examples are shown below:
 0.000000002 ÷ 10^{1} = 0.000000002 × 10^{1}
 0.00000002 ÷ 10^{2} = 0.00000002 × 10^{2}
 0.0000002 ÷ 10^{3} = 0.0000002 × 10^{3}
 2 ÷ 10^{10} = 2 × 10^{10}
So, using scientific notation (exponential notation), the diameter of an atom is 2 × 10^{10} metres.
The coefficient is 2
The exponent is 10
To summarise, scientific notation uses a small number (between 0 and 10) as the coefficient and multiplies that number by 10 raised to the power of a number called the exponent.
coefficient × 10^{exponent
}
If the number is greater than or equal to 10, the exponent will be positive (but we don't write the + sign)
coefficient × 10^{exponent
}
If the number is between 1 and 10, the exponent will be zero (0)
coefficient × 10^{0
}
If the number is between 0 and 1, the exponent will be negative (we DO write the  sign before the value of the exponent)
coefficient × 10^{exponent
}
Sometimes you might need to convert a number given in scientific notation (exponential notation) to a decimal system number.
For example, you might need to weigh out 1.325 × 10^{3} grams of table salt (sodium chloride).
The digital display on your electronic balance only gives numbers in the decimal system, so you will need to convert 1.325 × 10^{3} to a decimal system number.
Because the exponent is positive (no minus sign before the value of the exponent) we know we need to multiply "1.325" by "3" "× 10"s, that is:
1.325 × 10 × 10 × 10 = 1325
In effect, we are moving the decimal point to the right. The number of places the decimal point moves is equal to the value of the exponent:
decimal point in coefficient 

↓ 




coefficient 
1 
. 
3 
2 
5 

move decimal point 3 places to the right 

↑ 
→ 1 
→ 2 
→ 3 
↓ 
decimal system number 
1 

3 
2 
5 
. 
1.325 × 10^{3} = 1325. = 1325
What if we needed to weigh out 7.6 × 10^{3} grams of table salt (sodium chloride)?
Because the exponent is negative (a minus sign before the value of the exponent) we know we need to divide 7.6 by 10^{3}, or divide 7.6 by "3" "× 10"s, that is:
7.6 × 10^{3} = 7.6 ÷ 10^{3} = 7.6 ÷ (10 × 10 × 10) = 7.6 ÷ 1000 = 0.0076
In this case, we are moving the decimal point in the coefficient to the left.
The number of places it moves is equal to the magnitude (size) of the exponent.
If there is no number in the position where the decimal point moves, we insert a "0"
decimal point in coefficient 





↓ 

coefficient 




7 
. 
6 
move decimal point 3 places to the left 

↓ 
← 3 
← 2 
← 1 
↑ 

insert zeroes where required 

. 
0 
0 
7 

6 
7.6 × 10^{3} = .0076 = 0.0076