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Hexadecimal System


Date: 02/15/98 at 14:49:40
From: Dave Rhodus
Subject: Hexadecimal

I know the binary system using base two, but I don't understand the 
hexadecimal system using base 16. Can you please show me?


Date: 02/15/98 at 16:19:10
From: Doctor Sam
Subject: Re: Hexadecimal

Dave,

If you know the binary number system, then you understand that it is a 
place-value system like ordinary decimal numbers. The two threes in 
343 (base 10) stand for different things, and so do the two ones in 
101 (base 2).

In hexadecimal, the place-values are all powers of 16:

  ... 16^3  16^2   16^1     1

So the number 952 in in hexadecimal notation means 9(16^2) + 5(16)+ 2  
which is the decimal number 2386. One of the advantages of hex 
notation is that you can name larger numbers with fewer digits than in 
decimal notation (it takes four digits to write 2386 in decimal and 
only three digits to write the same value in hexadecimal.)

One of the problems with hex notation, however, is that in order to 
get bigger values into fewer digits we need more digits.

In binary we need only 2 digits: 0 and 1.

In decimal we need ten digits:   0,1,2,3,4,5,6,7,8,9

In general, you need as many digits as the size of the base. So in 
hexadecimal we need sixteen different digits. If that isn't clear 
think about why we need ten digits in decimal notation. If we have 
nineteen things to count we begin counting with the digits: 1,2,3,...  
but as soon as we get ten things, instead of using a new digit after 9 
we re-use the 1 in a new place value and write 10.

If we begin to count the same nineteen objects using hexadecimal 
notation, we get to the first two-digit number:  1 0 when we have 
counted sixteen objects (1(16)+0(1) = 16 ). That means that we have to 
count the first fifteen objects with only a single digit.

So we need some new digits!  The standard choice is to use the first 
six letters of the alphabet to stand for these numbers. Start as 
usual with 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 but after 9 write A to stand 
for ten, B for eleven, C for twelve, D for thirteen, E for fourteen, 
and F for fifteen. (Of course we don't need G for sixteen because that 
is 1 sixteen and 0 ones.)

For example, the hexadecimal number BC means B(sixteens) + C (ones) or    
11(16) + 12(1) = 188 in decimal.

Does that help?

-Doctor Sam,  The Math Forum
 Check our our Web site  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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