Number Bases, Grades 6-7 Page 2
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BLACKSBURG MATH CIRCLE
Number Systems of Arbitrary Base
So far we have seen two different number systems of different bases: decimal (base 10) and binary
(base 2). Now let’s think about the general idea of a number system with base
. What was
special about the numbers 10 and 2 in the two number systems we studied? There were a couple
of important things:
The place values were powers of the base.
The digits were allowed to be integers from 0 to the base minus 1.
So just by changing the base of a number system, we can change the way in which numbers are
represented completely. The base characterizes the number system. This motivates the following
definition: A base
number system is a number system in which every place value corresponds to
a power of
and a digit may only be an integer from 0 to
1.
For example, consider the base 6 number system. Place values correspond to powers of 6 and
digits may be integers from 0 to 6
1 = 5.
Problem 3. The Mayans used a base 20 number system. How do you think they learned to count?
Problem 4. What is the largest 3-digit number in the base system 6 is equal to? What is the largest
3-digit number in the base system
is equal to?
Problem 5. Can we use the base 1 system?
Problem 6. Let’s add binary numbers! Do the following binary calculations by first converting the
numbers to base 10, performing the calculations, and then converting the answer back to
binary:
0 + 0; 0 + 1; 1 + 0; 1 + 1; 1 + 1 + 1
Problem 7. Use the results from the previous question to do the following binary calculations without
converting to base 10. Check your answers by converting the numbers to base 10 and then
performing the calculations.
10
+ 1
;
1001
+ 110
;
111
+ 11
2
2
2
2
2
2
Hint: Think of place value and carrying (as with base 10 addition).
Problem 8. A palindrome is a positive integer whose digits are the same when read forwards or back-
wards. For example, 2002 is a palindrome. How many more 3-digit palindromes exist in
the decimal number system than in the binary number system?
Problem 7. What is the minimum number of weights which enables us to weigh any integer number of
grams of gold from 1 to 100 on a standard balance with two pans? Weights may be placed
only on the left pan.
Problem 8. The same question as in the previous problem, but the weights can be placed on either pan
of the balance.
Problem 9.
(a) How many different 5-digit binary numbers are there?
(b) How many different 5-digit binary numbers are there that have 1 as the last digit?
(c) How many different 5-digit base
numbers are there for a given
? Assume
2
10.
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