An approach to binary numbers

Binary numbers are intimately associated with the concept of parity, odd and evenness.

We all notice pretty quickly that 11111110 is even and 00000001 is odd.

The least significant bit, lsb, determines the overall parity of the number.

What is often overlooked is that parity pervades every bit of the number.

Take the binary representation 0101 of the decimal number 5 for instance.

The binary notation tells us immediately that 5 contains an:-

odd number of 1s

even number of 2s

odd number of 4s

even number of 8s

since odd parity is represented by a 1 and even with a 0.

Lets use this to convert 13_decimal to binary.

13 contains

one 8 ie an odd number of 8s

three 4s ie an odd number of 4s

six 2s ie an even number of 2s

thirteen 1s ie an odd number of 1s.

Therefore 13_decimal = 1101_binary

Without recognizing the parity hidden in every bit of a binary number the following method of converting decimal to binary might seem like magic. Perhaps it still does.

To convert the decimal number 34 to binary write down 34. Then

Write down half of 34.

Then halve that.

Discard any remainders eg half of 17 is 8 discarding the remainder.

Inevitably a 1 is reached.

Halving 1 gives 0 as the remainder is discarded.
Halving 0 gives 0 and no further change is possible.

Halvings	Parity
34	0
17	1
8	0
4	0
2	0
1	1
0	0
0	0

Every binary number starts with a 1 unless place holding leading zeroes are required to make up the bit total to, say, eight bit binary.

Also, 34 is an even number so clearly the binary representation must end in a 0.

34 contains an odd number of 2s since 17 2 so the second to least significant bit is 1

All in all it is obvious the first bit in the column gives the parity of the overall number so is the lsb and the most significant bit, msb is at the bottom of the column.

So rotate the parity column a quarter turn to the right and there is the binary number.

Best wishes.

Have fun and let me know how easy this is to read.

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