Binary number system
The main difference between the binary number system and our familiar base 10 numeration system is that
grouping is done in groups of 2 instead of 10.
For instance, to represent 24 in base 10 using sticks, you could use two groups of ten and 4 as shown below
There is something important though to keep in mind and this is the key to understanding this lesson!
- The digits 0,1,2,3,4,5,6,7,8,9 are used to represent all possible numbers. Notice that base 10 has 10 digits
- Depend on how big the number is, we make groups of ten, hundred, thousand, ten-thousand, etc...(These are power of 10: 10^{1} = 10, 10^{2} = 100, 10^{3} = 1000)
- If a number is less than 10 for example 8 and 9, there is no need to create groups. And this number will occupy the ones place value
- If a number is bigger than 9 and less than 100 for example 10, 55 and 98, there is a need to create groups of ten. Groups of ten will occupy the tens place value.
- If a number is bigger than 99 and less than 1000 for example 100, 255 and 999, there is a need to create groups of hundred. Groups of hundred will occupy the hundreds place value.
- And so forth...
For example, carefully study the following number to see how it was organized.
Since the number is bigger than 99, we had to make groups of hundred and ten. Notice also how groups of hundred are put in the hundreds place value and groups of ten are put in the tens place value.
In a similar way, the binary number system has its own place value.
- The digits 0,1 are used to represent all possible numbers in the binary number system. Notice that base 2 has 2 digits to represent all possible numbers.
- Depend on how big the number is, we make groups of 2, 4, 8, 16, 32 etc...(These are power of 2: 2^{1} = 2, 2^{2} = 4, 2^{3} = 8)
- If a number is less than 2 for example 1 there is no need to create groups. And this number will occupy the first place value. This place value correspond to the ones place in base 10
- In fact this 1 is the same in the binary system and the base 10 system.
- If a number is bigger than 1 and less than 4 for example 2 and 3, there is a need to create a group of two. A group of two will occupy the second place value. You could also call it "two" place value
- If a number is bigger than 3 and less than 8 for example 4 and 7,there is a need to create a group of four. A group of four will occupy the third place value. You could also call it "four" place value
- If a number is bigger than 7 and less than 16 for example 8, 11, and 14, there is a need to create a group of eight.A group of eight will occupy the fourth place value. You could also call it "eight" place value.
- If a number is bigger than 15 and less than 32 for example 16, 21, and 30, there is a need to create a group of sixteen. A group of sixteen will occupy the fifth place value. You could also call it "sixteen" place value.
- If a number is bigger than 31 and less than 64 for example 32, 45, and 63, there is a need to create a group of thirty-two. A group of thirty-two will occupy the sixth place value. You could also call it "thirty-two" place value.
- And so forth...
Above, notice the use of "A" repeatedly in " A group of..."
Well, this means that we can only create 1 group not 2, not 3
Since we are only using 0 and 1, to represent numbers, it will not be possible to write binary numbers using the number 2.
Now let us convert 25 to the binary number system
Notice that there are no groupings of two and four. As a result, put zeros in the second place and the third place
How grouping is done in the binary number system. How to convert from base 10 to base 2
Group from the nth place to the first place. This means that you have to create first a group with the highest possible power of 2
For instance, I am trying to convert 45 to a binary number system.
Ask yourself. What is the highest power of 2 less than 45?
2
^{6} = 64. 2
^{5} = 32. So, the highest power of 2 less than 45 is 2
^{5} = 32
Since 32 goes in the sixth place, put 1 in the
sixth place. Then try to find out what goes into the place value before.
45-32 = 13
Ask yourself. What is the highest power of 2 less than 13?
2
^{4} = 16. 2
^{3} = 8. So, the highest power of 2 less than 13 is 2
^{3} = 8
Since 8 goes in the fourth place, put 1 in the
fourth place. Then try to find out what goes into the place value before.
13-8= 5
Ask yourself. What is the highest power of 2 less than 5?
2
^{3} = 8. 2
^{2} = 4. So, the highest power of 2 less than 5 is 2
^{2} = 4
Since 4 goes in the third place, put 1 in the
third place. Then try to find out what goes into the place value before.
5-4 = 1 and 1 goes into the
first place.
Notice that you put nothing into the fifth place and the second place, so zeros go in those places
Therefore, 45 converted to the binary number system is 101101
You can also write 45
_{ten} or 101101
_{two}
Be very careful when you read 101101
_{two}!
101101
_{2} is read one zero one one zero one base 2