

6.3 Standard Notation
Hopefully you now know that exponential forms and power of tens are a good way of writing large or very small numbers. Another way of writing these is called standard notation. This is a simplification and the numbers is written in the form of a product. You could say that we build upon and develop the power of ten form for those numbers that aren’t even 10, 100, 1,000, 10,000 and so forth.
Here is an example of a power of ten form and standard notation.

Foto: José Raúl Vidal 
In Sweden there were about 100,000 children born in 2005. How can we write this in an exponential form. In section 6.2 you learned how to write 100,000 as a power of ten.
100,000 = 10 · 10 · 10 · 10 · 10 = 10^{5}
There were born in fact 10^{5} children in Sweden in 2005.
The number 10^{5 }is expressed as a power of ten. 

At the same time about 90,000 people in Sweden died in 2005.
If we write the number 90,000 in standard notation we can think as follows:
90,000
= 9 · 10,000
Then it becomes easier to see that: 
Foto: Hogia AB,
Multimediabyrån 
90,000 = 9 · 10,000 = 9 · 10 · 10 · 10 · 10 = 9 · 10^{4}
There died about
9 · 10^{4} people in Sweden in 2005.
The number 9 · 10^{4 }is written in standard notation.



It becomes a little more difficult if it turned out that 93,000 people died in Sweden. How should we write 93,000 in standard notation?
93,000 could be written as:
93 · 1,000 = 93 · 10 · 10 · 10 · 10 = 93 · 10^{3}
It is entirely correct, but a number in standard notation should always be expressed as a number between 1 and 10 multiplied by a power of ten.
How can we write the number 93 as a number between 1 and 10 without changing its value?


93,000 = 9.3 · 10,000 = 9.3 · 10 · 10 · 10 · 10 = 9.3 · 10^{4}
9.3 · 10^{4} is read as “Nine point three times ten raised to the four”.
The factor, which you multiply the power of ten with, must always be a number greater than or equal to 1 or less than 10 when you write a number in standard notation. This is something mathematicians have dictated! 

Can you write a number which is less than 1 in standard notation?
How do you think for example you write 2.3 · 10^{2}?
First we look at what the 10^{2} stands for. It corresponds to 1/100 = 0.01
So the number corresponds to 2.3 · 10^{2} = 2.3 · 0.01 = 0.023
This is a number which is less than 1 written in standard notation. So it is possible to write a number less than 1 in standard notation. 


