## Ask Professor Puzzler

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*"Sir , I saw your post about how to convert from alphabet to binary , in which I agree about the post. But there was a question I saw I'm my textbook which state that convert the hexadecimal number 4B3.3 to decimal. In your post here https://www.theproblemsite.com/ask/2016/01/converting-letters-to-binary, I notice that A= 65 and B =66. then I equate 66 to represent B. but after solving it there answer is different from mine .they got 1203.1875, and I got 18019.1875. How come sir? But I notice they equate there B = 11. Sir, I need your highlight on it. Thanks for talking time to read it, waiting for ur reply."*

Hi Abolade, Thanks for asking this question. To understand the answer to that, I need to talk for just a moment about graphemes. A grapheme, according to one dictionary, is "The smallest meaningful contrastive unit in a writing system." That's a fancy way of saying that a grapheme is a symbol that represents something meaningful in a writing system. For example, "7" is a grapheme for the number seven. It's a symbol, and whenever you see it, you automatically know that it represents this many things:

X X X X X X X.

On the other hand, "72" is not a grapheme because it's not the *smallest* graphical unit. "72" is actually two graphemes: the grapheme for the number seven, and the grapheme for the number two.

Graphemes are also used to represent letters. "A" is the grapheme we use to represent the first vowel in the alphabet, "B" is the grapheme we use to represent the first consonant, and so forth. Of course, the graphemes for these letters might look different if you were writing a different language, such as Greek: α β γ δ ε...

So here is where things get confusing. In base ten (our normal counting system), we have ten digits. And therefore, we have ten graphemes: 0 1 2 3 4 5 6 7 8 9. Perfect! We have just enough numerical graphemes! And if we're in base eight (also known as "octal"), we have eight graphemes: 0 1 2 3 4 5 6 7. We have more than enough graphemes (the graphemes for eight and nine don't get used in this base).

In fact, for any base less than ten, we have more than enough graphemes. The problem is when we start talking about bases *greater* than ten. Then we don't have enough graphemes!

So instead of inventing *new *graphemes to represent these digits, mathematicians said, "Why waste the effort developing new symbols, when we've got all these other graphemes lying around not being used?" Specifically, we're talking about the alphabet graphemes. So if you are in base eleven, your graphemes are: 0 1 2 3 4 5 6 7 8 9 A, where A is the grapheme for the number ten.

Similarly, if you are in base sixteen (also known as "hexadecimal"), you have sixteen graphemes, and six of them are stolen from the alphabet: 0 1 2 3 4 5 6 7 8 9 A B C D E F.

It's very important to note that in this context, A, B, C, D, E, and F, even though they look like the graphemes for LETTERS, have a very different meaning; they are now graphemes for NUMBERS!

A is the grapheme for the number ten

B is the grapheme for the number eleven

C is the grapheme for the number twelve

D is the grapheme for the number thirteen

E is the grapheme for the number fourteen

F is the grapheme for the number fifteen

The mathematicians very selfishly pirated letters from the alphabet to take on a completely different meaning. I mean, why not? It's not like there would be any confusion, right? Shakespeare was never going to write a sonnet about the hexadecimal number system, so there's no chance there would ever have an overlap of meaning where we were unsure whether A represents a number or a letter, right?

Wrong! Welcome to the world of computers! In a computer system, even though you can *type* graphemes and a computer can *display* graphemes, the computer does not *understand *graphemes. Computers "think" strictly in terms of numbers, not graphemes. Which means that people who designed computers had to come up with a system of converting graphemes into numbers, so the computer would be able to handle them.

How many graphemes are there? Well, don't forget that not only are there number graphemes and letter graphemes, there are also punctuation graphemes, mathematical operation graphemes, and special symbols like the pipe character, the tilde, etc.

So somewhere along the way, someone (or a committee of someones) had to develop a conversion chart so that whenever someone typed a character on the keyboard, there was a *number* that corresponded to whatever they typed. For example:

33 is the number that represents the exclamation mark grapheme

34 is the number that represents the quotation mark grapheme

43 is the number that represents the addition grapheme

65 is the number that represents the upper case "A" grapheme

66 is the number that represents the upper case "B" grapheme

97 is the number that represents the lower case "a" grapheme

Uh oh...now we have a problem! The grapheme "A" is a representation for the number 10, but 65 is the *computer's* value for the "A" grapheme. And this is the source of the confusion. Consider the following:

BAD = a hexadecimal number

BAD = of poor quality or low standard

Both of these statements are true. BAD is a hexadecimal *number*, but it's also a *word!* In one case, B, A, and D are graphemes for numbers, and in the other case, they are graphemes for letters! If you wanted to convert BAD (the number) into base ten, you would do this: 11 x 16^{2} + 10 x 16 + 4 = 2980. But if you wanted to convert BAD (the text) into numbers, so the computer could deal with it, you would do this: B = 66, A = 65, D = 68.

So if you're dealing with numbers (as in your textbook), A = 10, B = 11, etc. If you're dealing with text (words), A = 65, B = 66, etc.

If only those lazy mathematicians had invented their own graphemes instead of just stealing from the alphabet, we wouldn't have this confusion!

Thanks for asking, and I hope there was something helpful to you in this lengthy explanation!

Professor Puzzler

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