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!
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.
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?
26 = 64. 25 = 32. So, the highest power of 2 less than 45 is 25 = 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?
24 = 16. 23 = 8. So, the highest power of 2 less than 13 is 23 = 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.
Ask yourself. What is the highest power of 2 less than 5?
23 = 8. 22 = 4. So, the highest power of 2 less than 5 is 22 = 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 45ten or 101101two
Be very careful when you read 101101two! 1011012 is read one zero one one zero one base 2
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