Currently IEEE-754 is being used as standard for storing decimal numbers (or rather say float point numbers) in memory. Refer <a href="https://en.wikipedia.org/wiki/Floating_point" target="_blank" rel="nofollow noopener noreferrer">Floating Point</a> for more details on floating point number and its binary representation.

-Pradeep

I'll give a part of an older article I wrote on a website about subnetting, for teaching you how to fish instead of giving you a kg of fish 😀. This is yet another method that works best on small decimal numbers (less then 1 byte usually).

A) Binary to decimal conversion

To fully understand these kind of transformations, you should firstly know these values:

2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
(will refer to this as Table A.)

Now how do we transform binary to decimal?

Let's take the number 10100110 which is in binary (ones and zeros) and transform it to decimal.

We just take all values and form a table, starting with 2^0 (lowest value) from the right position of our binary number. (you'll do this only the first times, after some practice you'll do it mentally with no problems):

1 0 1 0 0 1 1 0
2^7 2^6 2^5 2^4 2^3 2^2 2^1 2^0

Now we count only the values with a one (1) above and calculate their sum.
The result is 2^7+2^5+2^2+2^1=(using Table A.)=128+32+4+2=166
So 10100110 (binary) = 166 (decimal).


B) Decimal to binary conversion

For this transformation you also need the values in Table A.

Let's take for example the number 240 and try to find out the binary equivalent.
The main idea behind this is to try to write the decimal number as a sum of numbers in Table A. (exponential powers of 2).
The easiest way is this:

Find the highest number that can be written as a power of 2 that is smaller or equal then our decimal number (from Table A).

256 is the highest, but its bigger than our number (240) so it's not good. Next is 128 which matches our criteria.

Subtract 128 from 240: 240 - 128 = 112
Note the number 128 down.

Now do the same for 112. Find the highest number in Table A smaller then 112. This value will be 64.
Subtract: 112 - 64 = 48.
Note the number 64 down.

Find the highest value smaller than 48 in Table A. It's 32.
Subtract: 48 - 32 = 16
Note 32 down.

Now 16 is a an exponential power of two, it's the highest value in Table A smaller or equal to 16 so we stop. Note 16 down too.

Put all numbers noted down:
128+64+32+16=240 our number. We managed to write our decimal number as a sum of exponentials powers of 2.

Now write all values from 2^0 to 2^7 (one byte long data). Write the decimal equivalent of our noted down numbers, and put ones (1) under them. Put zeros under the numbers we didn't noted down.

2^7(128) 2^6(64) 2^5(32) 2^4(16) 2^3 2^2 2^1 2^0
1 1 1 1 0 0 0 0

The number in ones and zeros you obtains is the binary equivalent of the decimal.
SO 240 (=2^7+2^6+2^5+2^4) = 11110000.


Best regards,

Mihai


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