## Converting between binary, decimal, and hexadecimal notations

Ever since the Stone Age we humans have counted on our ten fingers. Our system of numbers shows it. We have ten elementary numbers called "digits"; a word that also means fingers. We count on our fingers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To go further, we use place notation, allowing us to represent larger numbers as sequences of digits. Each digit in such a sequence has a different value depending on its place. For example, 13 means ten x 1 + 3, a number 4 greater than 9. We usually call the number thirteen, an alteration of "three ten". When we run out of two-digit numbers at 99, we go to three digits, with 100, which means 1 x hundred + 0 x ten + 0. A hundred is ten times ten. Similarly, the number 1000, with a one in the fourth place left of the end, means ten times ten times ten, or ten times a hundred, or a thousand. Place notation allows us to represent any conceivable number as a sequence of digits.

This may be fine for human beings who twiddle fingers all the time, but the idea of tenness is unknown to computers. All they understand is on and off. Either a bit (a core in memory or a place with a charge) is charged or discharged; magnetized or unmagnetized, and so forth. That is just two values. Therefore, computers are built to be based on a number system where one counts by twos, the binary system.

In binary, there are just two digits, 0 and 1. We may have .jpg files, HTML browsers, Power Point presentations, and DLLs, but when you get down to the lowest layer of any of these, they turn out to be just strings of ones and zeroes. It turns out you don't need ten digits to count all conceivable numbers. All you need is just two digits, or bits (short for binary digits). So one can start counting with 0, 1. What's next? Use place notation. The next number can't be 2, since we are not allowing use of that digit. So we must go to the next place and call it 10. This is not ten. It is two. That says the one in the second place means 1 times two. We continue counting, and go 0, 1, 10, 11. Now we run out just like we did at 99 in decimal. So we go to the third place and get 100, which we call "four". So the third place represents fours. Each place is double the previous one, so the next one is eights, then sixteens, thirty-twos, and so forth.

Take a binary number like 10101011. This means 1 x 128 + 0 x 64 + 1 x 32 + 0 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1, which is 128+32+8+2+1 or 171 in decimal. (I used decimal numbers like "16" since that is just about the only way we know them).

After working with them a while, we can remember that 1100 is twelve and 101 is five and so forth, but what is 110100101010110101000100001111010101010100010101? I am not sure even by glancing at it how big that number is! Human beings are just not built for binary numbers. We can conceive of a concept with ten values, like a digit, but we can hold only 6 or 7 digits in our mind at any time. This limits us in binary to about 128.

To make it easier to comprehend computer numbers, another base is used: sixteen, which is two to the fourth power, or 10000 in binary; 16 in decimal. In this system we count by sixteens. So we count 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, like in decimal. We need more digits to get to sixteen. Human culture has not provided us with any digits, so we need to make up our own. Usually we use the other common type of character in our culture, the alphabet letter. So we continue A, B, C, D, E, F. This means that A is ten, B is eleven, and so forth to F which is fifteen. After that comes 10, meaning sixteen. Then 11, 12, ..., 100 (which is sixteen times sixteen or 256), and so forth.

We could do all our arithmetic in base sixteen. We would have to learn new addition and multiplication tables (4 + 8 = C; 7 x 9 = 3F, and so forth), and there would be more facts to learn. However, decimal is likely to remain supreme for humans in the near future. But why bring up base sixteen, also called hexadecimal, which gets Latin about it? Because it is easy to convert from binary to sixteen. It turns out that hexadecimal digits correspond to groups of four binary digits, so that 5 is 0101, B is 1011 and so forth. This means that 10110101 = 1011 0101 is B5 in hexadecimal.

Still if we use base 2 and sixteen for computers, there is the need for converting these between each other and decimal. It is not obvious that 171 is 10101011 in binary or AB in hexadecimal. We need a quick way of converting back and forth.

I find these methods to be quickest by hand. I will explain first how to go between decimal and binary. Then I will describe how to go between either of these and hexadecimal. Given a decimal number like 233. How do you convert it to binary? First write down 233 and write half the number below it (116). If there is a remainder, ignore it. Repeat the process; that is, write half of 116 below the 116. That is 58. Continue until you get to 1. The result is:

 233 116 58 29 14 7 3 1

Now write to the right of each number in this column a 1 if the number is odd and a 0 if the number is even. The result is:

 233 1 116 0 58 0 29 1 14 0 7 1 3 1 1 1

Now read off the binary representation, going up from the bottom, to get 11101001.

To convert a binary number to decimal, one takes this process in reverse. Suppose one wants to convert 101110101101 to decimal. Write the number in a column like this, and place a zero to the upper left of the top of the column:

 0 1 0 1 1 1 0 1 0 1 1 0 1

Starting with the 0, double it down the column. If the digit to the right is a 0, simply double the previous number. If it is a 1, double the previous number and add 1. Since the first digit is a 1, double the 0 and add 1. The result is 1. The next digit is a 0. So simply double the 1 to get 2. Since the next digit is a 1, double the 2 and add one to get 5. Keep doing this down the column and the last number in the column on the left will be the decimal number that corresponds to binary number 101110101101. The result is:

 0 1 1 2 0 5 1 11 1 23 1 46 0 93 1 186 0 373 1 747 1 1494 0 2989 1

So the decimal equivalent is 2,989.

Now how about hexadecimal? First of all, converting between binary and hexadecimal. The first thing to do is to memorize this table. Know it by heart. When you see 1010, automatically think "A". 0110 should bring you visions of "6". If you know Morse Code, it is the same sort of drill, but with only 16 digits instead of 26 letters. Memorize this table:

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

Once you have this table memorized, it is simple to convert from binary to hexadecimal. Simply split the binary number into groups of four and translate them into hexadecimal digits that you just memorized. For example, to convert 101110101101 to hexadecimal, split it into fours:

1011 1010 1101
Then translate the fours into hexadecimal:
 1011 1010 1101 B A D

so the hexadecimal for 101110101101 is BAD. No that does not mean there is something wrong with it. BAD is the hexadecimal number "eleven ten thirteen".

To convert from hexadecimal to binary, do the process in reverse. Replace each hexadecimal number with its binary equivalent. For example, let's ring out the old DECADE. No, let's just convert it to binary for now. The result is:

 D E C A D E 1101 1110 1100 1010 1101 1110

so the binary is 110111101100101011011110.

To convert between decimal and hexadecimal, convert to binary first. For example, to convert 1001, a thousand and one, to hexadecimal, first convert 1001 to binary:

 1001 1 500 0 250 0 125 1 62 0 31 1 15 1 7 1 3 1 1 1

The binary is 1111101001. This has ten digits, which is not divisible by four, so add zeroes to make it divisible by four. In this case, add two zeroes: 001111101001. Then split it up and obtain the hexadecimal:

 0011 1110 1001 3 E 9

To convert the other way, from hexadecimal, convert it to binary and then to decimal. For example, to convert 2B2B to decimal, convert it to binary first:

 2 B 2 B 0010 1011 0010 1011

The binary for 2B2B is 0010101100101011, or 10101100101011, since the initial zeroes don't add anything. Convert it to decimal:

 0 1 1 2 0 5 1 10 0 21 1 43 1 86 0 172 0 345 1 690 0 1381 1 2762 0 5525 1 11051 1

The decimal for 2B2B is 11,051.

With these methods in mind, once you memorize that table above, you can quickly convert from decimal to binary and back again. Good luck converting!

Jim Blowers

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