what is the purpose of number system conversions e.g decimal to base 5?

Question

I'm learning Number system conversion youtube. so far I know there are decimal, binary, octal, and Hex numbers. There is a purpose behind converting decimal to binary But what is the purpose of converting decimal to base 5 number?

Answer 1

Not much. The only argument for base $10$ is that we are used to it, which is very strong. Binary is useful for computer processing, but leads to very long expressions for numbers. Years ago some computers would compress binary into octal, which cuts the number of digits by a factor $3$. I worked on a CDC6400 that had $60$ bit words and the dumps were in octal. I haven't heard of that for a long time, now it is all hex, which cuts the number of digits by a factor $4$, but I am out of touch and there could be some systems that still use octal.

Long ago I did a math puzzle that had $6$ everywhere in layers. There was a bunch of computation to do, which I did in base $6$. The puzzle could be solved without that.

I think there is value in recognizing the distinction between a number and its representation in some particular way. Expressed in binary, $\frac 15$ does not terminate, while it does in base $10$. We get a number of questions involving terminating decimals that do not realize it depends on what base you are in.

Answer 2

Any choice of base is fairly arbitrary. Since we have 10 fingers on our hands, we use base 10. But we can use any other bases; lets explore some situations where they come up.

One example is that computers are very good at representing things in 2 positions. If we call one of those positions 0 and the other 1, we've suddenly imposed a base 2 system onto it, which we call binary.

Another example is that you can use your hands to count in base 6. On each hand, you can show a number of fingers between 0 and 5, so I can represent any number between 0 and 35 by multiplying the number of fingers shown on my left hand by 6, and adding the number of fingers on my right hand.

You could adapt the above system to base 5, for instance by ignoring your thumbs and instead multiplying the number of fingers on your left hand by 5 instead of 6. But this would limit the maximum number to 24, and it doesn't make as much sense.

So, what's the use of base 5? Knowing that its a tool in the toolbox, if a situation should arise where it's useful.

Answer 3

All the digital systems in this world works on 0 and 1, so in order to understand its working we should understand binary numbers, Also since currently all processors uses atleast 4 bit so hexadecimal is preferred in programming.