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How Does Base 4 Work?

Date: 11/06/2003 at 10:27:49
From: Jody
Subject: Base 4

How does Base 4 work?


Date: 11/06/2003 at 14:42:02
From: Doctor Riz
Subject: Re: Base 4

Hi Jody -

Bases have to do with how you write numbers in a number system, and
how the place values work in that system.  

Let's start with the system you already know.  We usually work in base 
10.  In base 10, the place values are ones, tens, hundreds, thousands 
and so on.  So when we see a number like 437, it really means 'four 
hundreds, 3 tens and 7 ones.'  We understand that to be worth 'four 
hundred thirty seven'.

The place values are determined by raising the base to powers.  In
base 10, ones is 10^0, tens is 10^1, hundreds is 10^2, thousands is
10^3 and so on.

When we start to count in base 10, we can write 

  1, 2, 3, 4, 5, 6, 7, 8, 9.  

Each of those stands for how many ones we have.  The number 8 means 
8 ones, or 8 * 1.  But when we go past 9 to the number 10, we don't
have a single digit that stands for '10 ones.'  So instead, we use a
two-digit number, 10, which stands for '1 ten and 0 ones.'  Once we
get to 99, we have reached '9 tens and 9 ones.'

Going past that, we move to a three-digit number, 100, which means '1
hundred, 0 tens and 0 ones.'

It's kind of hard to think about this, because your brain just does it 
without thinking about it, but that's what's really going on.

So what happens in base 4?  The place values are again given by
raising 4 to powers.  

  4^0 =  1 
  4^1 =  4
  4^2 = 16
  4^3 = 64

So, the number 23 in base 4 is NOT worth twenty-three.  It's only
twenty three in base 10, where it means '2 tens and 3 ones.'  In base
4, 23 (which is read as 'two-three') means '2 fours and 3 ones.' So it
has a value of 2*4 + 3*1 or 8 + 3 or 11.

Now think about how we count in base 4.  We start with 1, 2, 3.  But 
there is no digit '4' to use--the number 4 is written '1 four and 0 
ones,' so it's 10.

I know this may be confusing, but here are the numbers from 1 to 10 in 
base 4:

 Base 4              Meaning             Base 10
 ------  ------------------------------  -------
   1     (1 one)              =     1*1     1
   2     (2 ones)             =     2*1     2
   3     (3 ones)             =     3*1     3
  10     (1 four and 0 ones)  = 1*4 + 0     4
  11     (1 four and 1 one)   = 1*4 + 1     5
  12     (1 four and 2 ones)  = 1*4 + 2     6
  13     (1 four and 3 ones)  = 1*4 + 3     7
  20     (2 fours and 0 ones) = 2*4 + 0     8
  21     (2 fours and 1 one)  = 2*4 + 1     9
  22     (2 fours and 2 ones) = 2*4 + 2    10

One more example.  What would 312 in base 4 be worth?  Since the third 
place value is 4^2 or sixteens, 312 means '3 sixteens and 1 four and 2 
ones,' so it's worth 3*16 + 1*4 + 2*1 or 48 + 4 + 2 or 54.

Notice that each place value 'spills over' into the next one just in 
time.  For example, the biggest two-digit number we can write in base
4 is 33, which is worth '3 fours and 3 ones.'  That would be worth 15.  
To get to 16, we move into the next place value and use a 3 digit 
number, 100.  That means '1 sixteen, 0 fours and 0 ones.'

Notice also that in base 4 the largest single digit is 3, just as in 
base 10 the largest single digit is 9.  Sometimes people work in a
base larger than 10, and in order to do that you need to make up new
digits.  

In base 12, for instance, the first place value is 12^0 or ones and
the next place value is 12^1 or twelves.  That means when you count
and get up to 9, you need some single digit to represent 'ten' and
another for 'eleven' since you don't spill over into the twelves place
until you get to twelve and write it as 10 (1 twelve and 0 ones).  

Many times things like * or # are used as a single digit to represent
having 10 or 11 of something.  Some graphing calculators can work in
base 16, which is called 'hexadecimal.'  In hexadecimal, we use
letters of the alphabet for additional digits, 

   Base 10:  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16
   
   Base 16:  1  2  3  4  5  6  7  8  9   a  b  c  d  e  f 10

Base 2 is a commonly used base because it only has two digits, 0 and 
1.  Computers do most of their work in base 2, because a computer has 
lots of tiny electronic switches, and each one can be either off (a 0) 
or on (a 1).  Base 2 is referred to as 'binary.'

This may have been more than you wanted to know, but I wanted to try
and explain it all.  Bases are very confusing to most people, but they 
don't need to be.  The trick is to really think about how our base 10 
number system works--all the same ideas work in other bases, they
just have a different set of digits, which leads to different place
values in the numbers.

Hope that helps--write back if you are still confused.

- Doctor Riz, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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