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

I know the binary system using base two, but I don't understand the
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Date: 02/15/98 at 16:19:10
From: Doctor Sam

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/
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Associated Topics:
High School Number Theory

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