

7.1 Different Number Bases In today’s mathematics, we use a position system with a base of 10 which is called the decimal system. The numbers in the decimal system are built with the help of our 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
These digits are put together to form numbers and depending on which place a digit has in the number, the value will be different.
Every digit in a number in the decimal system corresponds to a power of ten. Its place value is determined by the size of the power of ten.
It become clearer if we write a number in another form.
8,349 = 
8 · 1,000 
+ 
3 · 100 
+ 
4 · 10 
+ 
9 · 1 
8,349 = 
8 · 10^{3} 
+ 
3 · 10^{2} 
+ 
4 · 10^{1} 
+ 
9 · 10^{0} 
The Binary Number System
Maybe you have heard about the binary number system? We commonly say that a computer only contains ones and zeroes. What we mean is that the computer does all its calculations with the digits 1 and 0. It uses the binary number system. The binary number system is also a position system but instead of base 10, it has the base 2.
Every digit in a number in the binary number system corresponds to a certain power of 2.


Example 1
What value does the number 1,101_{two} have, if we write it in the decimal system?
We can write this in another form just as we did in the example before:
1,101_{two}= 
+ 
1 · 2^{3} 
+ 
1 · 2^{2} 
+ 
0 · 2^{1} 
+ 
1 · 2^{0} 

1,101_{two}= 
+ 
1 · 8 
+ 
1 · 4 
+ 
0 · 2 
+ 
1 · 1 

1,101_{two}= 
+ 
8 
+ 
4 
+ 
0 
+ 
1 
= 13_{ten} 
1,101_{two}= 
13_{ten} 







Example 2
What does the number 29_{ten} have, if we write it in the decimal system?
We can write this in another form just as we did in the example before:

2^{5} 
2^{4} 
2^{3} 
2^{2} 
2^{1} 
2^{0} 


32 
16 
8 
4 
2 
1 

29_{ten }= 

1 · 16 
1 · 8 
1 · 4 
0 · 8 
1 · 1 

29_{ten }= 

1 
1 
1 
0 
1 
= 11,101_{two} 
29_{ten }= 11,101_{two}
Five Base System
Why did we choose the ten base as our system of number? It might be because we have just 10 fingers. A useful method to find out how many things we own is to count on our fingers. Imagine if people only had one arm and because of this, only five fingers. We might instead have had a five base system.
The numbers in a five base system are built with the digits: 0, 1, 2, 3, and 4. Every digit in a number in the five base system corresponds to a power of 5.
Example 1
What value does the number 2,341_{five }have if we write it in the decimal system?
2,341_{five} = 
+ 
2 · 5^{3} 
+ 
3 · 5^{2} 
+ 
4 · 5^{1} 
+ 
1 · 5^{0} 

2,341_{five} = 
+ 
2 · 125 
+ 
3 · 25 
+ 
4 · 5 
+ 
1 · 1 

2,341_{five} = 
+ 
250 
+ 
75 
+ 
20 
+ 
1 
= 346_{ten} 
2,341_{five} = 
346_{ten} 







Example 2
What does the number 4,573_{ten }have if we write it in the decimal system?

5^{5} 
5^{4} 
5^{3} 
5^{2} 
5^{1} 
5^{0} 


3,125 
625 
125 
25 
5 
1 

4,573_{ten }= 
1 · 3,125 
2 · 625 
1 · 125 
2 · 25 
4 · 5 
3 · 1 

4,573_{ten }= 
1 
2 
1 
2 
4 
3 
= 121,243_{five} 
4,573_{ten } = 121,243_{five}


