Intro to other bases
From Compsci1
Introduction to other bases.
Write down all the digits you know.
There are 10 digits. This is called base 10 or denary. Does this mean that we can only count as far as the number of digits. What happens if we want to count the following • * * * * * * * * * *
What happens if we want to count
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
- * * * * * * * * *
What do the digits in a number such as
1023 actually represent?
The 3 is units
The 2 is tens
The 0 is hundreds
The 1 is thousands
1023 = 1x 1000 + 0x100 + 2x10 + 3x1
= 1 x 103 + 0 x 102 + 2 x 101 + 3 x 100
What happens if we only have 2 digits, 0 and 1.
Count in this code:
0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, etc.
This code or base which only has 2 digits – 0 and 1 – is called binary code and is the language of a computer.
If we look at binary numbers in the same way we look at base 10, then a number such as 111 = 1 x 22 + 1 x 21 + 1 x 20
= 1 x 4 + 1 x 2 + 1 x 1
= 4 + 2 + 1 = 7 in base 10.
An easy way to change a number from base 10 to base 2 is to divide repeatedly by 2, writing out the
remainer each time as follows:
To change 23 from base 10 to base 2
2 23
2 11 + 1
2 05 + 1
2 02 + 1
2 01 + 1
2 00 + 1
If we read the remainders from the bottom up the answer is 2310 = 111112
Try changing the following numbers from base 10 to base 2 using the above method
8
12
15
19
39
45
27
12
63
64
65
124
126
127
128
129
299
What patterns do we see emerge?
If the decimal number is odd the binary number will end in 1.
If the decimal number is even the binary number will end in 0.
As the decimal numbers get bigger the binary numbers become very long.
