Number Bases, Grades 6-7
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2NUMBER BASES, GRADES 6-7, SESSION 1
BLACKSBURG MATH CIRCLE
The Decimal System
We learn to count in only one of many possible number systems: the decimal system. The
decimal system is our default number system. The word decimal is derived from the Latin word
decem which means ten. Strictly speaking, the decimal system is a number system of base 10.
Don’t worry about all the fancy language right now. Just follow along with the example below to
find out what is so special about the number 10 in the decimal system.
Problem 1. (a) Write the following numbers in expanded form (as sums of powers of 10):
102079;
70000
6999
(b) Write the following number in standard form (the usual digit representation):
5
3
1
0
0
(9)(10
) + (2)(10
) + (8)(10
) + (1)(10
) + (4)(10
)
Binary Numbers
Another number system that is used every day is the binary system. It is the number system
that computers use. It can be described as follows: The binary system is a number system with
base 2. Every place value corresponds to a power of 2 and a digit may only be either 0 or 1. Some
examples of binary numbers are 11
, 10001
, 110011
, and 10101011
. Note that these are not read
2
2
2
2
as eleven, ten-thousand and one, etc., but as one-one, one-zero-zero-zero-one, and so on (to avoid
confusion). Just like the decimal system used powers of 10 to create place values, the binary system
uses powers of 2 to do the same. Look at the expanded form of the binary number 110110
and
2
the way it is written in the place value chart:
5
4
3
2
1
0
110110
(1)(2
) + (1)(2
) + (0)(2
) + (1)(2
) + (1)(2
) + (0)(2
)
2
5
4
3
2
1
0
Place Value 2
2
2
2
2
2
Place Value 32 16
8
4
2
1
Digit
1
1
0
1
1
0
Just as we did with the decimal numbers, we can now write 110110
as a sum of powers, in this
2
case powers of 2. So the binary number 110110
is the decimal number 54. Notice that all we ac-
2
tually need to do is add up the numbers in the Place Value row that have a 1 below in the Digits row.
Problem 2. Write the decimal number 238
as a binary number; write the binary number 110001
as
10
2
a decimal number.
Date: September 17, 2016.
1
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