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### Converting to Base 16; Place Value Chart

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Date: 03/22/98 at 16:13:09
From: Terri
Subject: Base 16 numbers

How do you convert numbers to base 16 numbers?

Please use the following numbers in an example: 411213 and 38015.

Thank you.
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Date: 03/22/98 at 20:01:39
From: Doctor Sam
Subject: Re: Base 16 numbers

Hi Terri,

Here's one way to think about other bases. If you were given 3
hundred dollar bills and 4 ten dollar bills and 2 one dollar bills
you would have 342 dollars.  We can just "glue" the 3 and the 4 and
the 2 together because our decimal number system is based on tens.

Now if I gave you 3 quarters and 4 nickels and 2 pennies you wouldn't
have 342 cents because nickels are only worth five pennies and
quarters are only worth five nickels. But in base 5 notation that is
just what you would write down. In base 5 notation every place value
is five times as large as the one before it.  So 3 quarters, 4
nickels, 2 pennies would be written as 342 base 5.

The key idea is that our system of writing numbers uses the idea of
place value: that digits written in different places mean different
amounts.

Here is a chart of some place values in different bases:

base 10:  ...    10*10*10      10*10    10    1

base 5:  ...    5 * 5 * 5     5 * 5     5    1

base 2:  ...    2 * 2 * 2     2 * 2     2    1

base 16:  ...    16*16*16      16*16    16    1

To change 411,213 begin by writing down the place values in base 16.
The first few are:

16^5       16^4       16^3       16^2      16^1      16^0
1048576    65536       4096        256       16         1

411,213 < 1048576 so we will not use that place. But 411213 > 65536 so
divide 65536 into 411213. The quotient is 6 and the remainder is 17997
so there will be a 6 in the 16^4 place.

Now work with 17997.  17997 divided by 4096 has quotient 4 and
remainder 1613 so there will be a 4 in the 16^3 place.

Now work with 1613.  1613 divided by 256 has quotient 6 and remainder
77 so there will be a 6 in the 16^2 place.

Now work with 77.  77 divided by 16 has quotient 4 and remainder 13 so
there is a 4 in the 16^1 place.

Finally, there is 13 remainder.  This is represented by a D in the
ones place.

Here's how to organize the work.

411213 = 6(17997) + 17997
17997 = 4(4096)  + 1613
1613 = 6(256)   + 77
77 = 4(16)    + 13
13 = D

Read down the quotients column to get the hex number 6464D.

To write this 38015 in hexadecimal do the same thing. Here's the work:

38015 = 9(4096) + 1151
1151 = 4(256)  + 127
127 = 7(16)   + 15
15 = F(1)

So 38015 = 947F.

Your examples don't illustrate the following possibility, which may
seem confusing.

To change 57536 to hex start again in the 4096 place (since 65536 is
too large).

57536 = 14(4096) + 192

Now 192 is smaller than the next hexadecimal place 256, but don't
ignore that place. It will get a zero. To finish the problem:

57536 = 14(4096) + 192
192 =  0(256)  + 192
192 = 12(16)  + 0
0 =  0 (1)

Now 14 = E and 12 = C so 57536 = E0C0.

I hope that helps.

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

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