# How number system works in cases of hex, bin, dec and oct?

I have this question or a confusion from college time. When we convert a decimal digit to binary, we divide the decimal digit with the base of binary number system. Why there is no similar method taught as hex to binary conversion. Suppose I have a number 0xAF, and its divided by 2. how can we see the remainder and quotient keeping all mathematical logic same as decimal.

• You can do it that way, but it's not done because it's not necessary and inefficient. That's because $16=2^4$ is a power of two, so each hexadecimal digit corresponds to $4$ binary digits (whereas there's no such correspondence between decimal and binary digits). Thus you just need to know the $16$ bit patterns of the $16$ hexadecimal digits to readily convert between hexadecimal and binary. – joriki May 19 '16 at 6:24

First of all, you can convert from hexadecimal to binary in the exact same manner:

0xAF/2 = 0x57 , 0xAF%2 = 1
0x57/2 = 0x2B , 0x57%2 = 1
0x2B/2 = 0x15 , 0x2B%2 = 1
0x15/2 = 0x0A , 0x15%2 = 1
0x0A/2 = 0x05 , 0x0A%2 = 0
0x05/2 = 0x02 , 0x05%2 = 1
0x02/2 = 0x01 , 0x02%2 = 0
0x01/2 = 0x00 , 0x01%2 = 1


Hence $AF_{16}=10101111_{2}$.

Second, due to the fact that $16=2^4$, a much simpler conversion method is at hand.

You can simply replace each hexadecimal digit, with a sequence of $4$ binary digits.

The conversion table is given below:

0 = 0000
1 = 0001
2 = 0010
3 = 0011
4 = 0100
5 = 0101
6 = 0110
7 = 0111
8 = 1000
9 = 1001
A = 1010
B = 1011
C = 1100
D = 1101
E = 1110
F = 1111

• how you divided 57/2 = 2B. Using calculator I can also do the same. But I want to see how we can do it numerically. – mrigendra May 19 '16 at 8:49
• @mrigendra: What do you mean "numerically"? You can use long division if you want, but it's not that easy when performed on a non-decimal base. You could convert first from hexadecimal to decimal, and then from decimal to binary... But again, there's a much simpler approach for this specific case. – barak manos May 19 '16 at 8:53
• numerically means, doing with hands with explanation on paper. – mrigendra May 19 '16 at 8:55
• @mrigendra: $AF_{16}=A\cdot16^1+F\cdot16^0=10\cdot16^1+15\cdot16^0=160+15=175$... $\lfloor175/2\rfloor=87$... $87_{10}=5\cdot16^1+7\cdot16^0=57_{16}$... – barak manos May 19 '16 at 9:03
• I got my answer. We do not memorize hex tables as we memorize decimal tables. Thats why I got confused in hex division. – mrigendra May 19 '16 at 9:26