Note: This page does not take into account Significant Figures. Please read and understand this article
first before reading about significant figures.
Scientific Notation
Scientific Notation is designed to simplify dealing with large numbers. For instance we can write
5,000,000,000 as 5 billion or as 5 × 109
When we multiply the number by 10 9 , we add nine zeroes after the number.
This works because we use a base 10 number system – every time we add a zero, the number gets larger
by factor of 10.
Most likely, we use a base 10 number system because we have 10 fingers – but that is
debatable. 1 It is interesting to note that the word digit means both a number from 0 to 9 or a
distal part of a limb, such as a finger.
Let’s look at how scientific notation represents smaller numbers
0.00015  1.5  104
Notice that after the decimal there are 3 zeroes and the leading number. That is four spaces
and typically that’s how I count scientific notation for decimals.
For example:
1.89  105  0.0000189
17,000  1.7  104
Try some conversions (answers are at the bottom of this post):
1. 17.6 
2. 0.00094 
3. 5,100 
4. 51 
5. 5.1 
6. 0.13 
7. 0.000000169 
8. 164  105 
With the last example we see something interesting:
164  105  0.00164 , we move the decimal 5 digits right due to 105 but because 164 is larger
than 1 by a factor of 100, we move it left 2 digits. There is an easier way I find of looking at this:
164  105  1.64  102  105  1.64  103  0.00164
We see that we can convert everything to scientific notation and then add and subtract the
exponents on base 10.
For example, 105  104  109 , this is the same as x 5  x 4  x 9
With like bases we can add the exponents when we multiply them:
44  48  412
24  28  212
104  108  1012
When we divide like bases, we subtract the exponents
24
1
 24  4
8
2
2
Notice that a negative exponent just places the number in the denominator.
104
1
 104  4  0.0001}
8
10
10
Notice that above 104  1  104  0.0001 (it is assumed there is a 1 in front of the base),
Let’s try multiplying some numbers with scientific notation:
5  104  3  103  15  107
Notice that all we did was multiply 5  3  15 and 104  103  107
This is because 5  3  4 is the same as 4  5  3 – the order of the operation does not matter!
This makes working with scientific notation a breeze. If you know some basic mental math you
can simplify large problems:
48  1015  9  105 6  1015  9  105 6  1015  9



8  105
1  105
1
54  1015  5.4  1016
In the first step, we divided 48 by 8. In the next step, we cancelled 105 from the numerator
and denominator. In the last step we multiplied 6 by 9 and simplified. The whole problem can
be done without a calculator.
It is recommended you do as much work as you can mentally – every time you take your pen off
the paper and switch to a calculator, you distract yourself from the work you are doing and
your work suffers. You must focus on the problems you are solving and build up your ability to
do small mathematical operations rapidly on paper with pen. When you do mathematical
operations in your head instead of the calculator you are exercising your brain. Learning mental
math will speed the rate at which you solve problems and make you think more rapidly overall.
Here are some reasons why (link to future article on mental math).
Let’s make it a little bit more complicated:
40  1015  9  105 40  1015  9  105 5  1015  3  105



24  1010
3  8  1010
1010
15  1010
 15
1010
In the first step, we noticed 24 has two factors, 3 and 8, that can be divided into 40 and 9 in the
numerator. This enabled us to simplify the mathematical operations we do. We then multiplied
the coefficients 3 and 5 and cancelled 1010 from the numerator and denominator. The whole
problem can be done without a calculator.
directly in our equation and achieved our result in a much more sensible unit of kilojoules or kJ.
Answers to Problem Questions:
1.
2.
3.
4.
5.
6.
7.
8.
17.6  1.76  101
0.00094  9.4  104
5,100  5.1  103
51  5.1  101
5.1  5.1
0.13  1.3  101
0.000000169  1.69  107
164  105  0.00164
References:
1. The Universal History of Numbers: From Prehistory to the Invention of the Computer,
Volume 1. Accessed September 14, 2013 at
http://books.google.ca/books?id=FMTI7rwevZcC&redir_esc=y
Useful Links:
Metric Conversions
http://www.convert-me.com/en/metric_conversions.html Accessed September 14, 2013
https://www.khanacademy.org/math/arithmetic/rates-and-ratios/metric-systemtutorial/v/converting-within-the-metric-system Accessed September 14, 2013
Scientific Notation
http://www.purplemath.com/modules/exponent3.htm. Accessed September 14, 2013
https://www.khanacademy.org/math/arithmetic/exponents-radicals/scientificnotation/v/scientific-notation-examples. Accessed September 14, 2013