Place Value and Numeration
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Other Bases and Numeration

In the Base Five lesson, we explored numeration by grouping in multiples of five. We could choose other bases as well, and construct some models that would work to represent various quantities. For example, we might use '8' as a base...

base eight pieces

We can go smaller, too...

base eight fraction pieces

We can then think about some base 8 notation for various quantities...

base eight model

What would the base 8 models and notation look like for the following quantities?

•1000 units? •741/2units? •1353/4 units?

We use a base two numeration system in computer applications. It's called the binary system. In fact, the term 'bit' means binary digit. Here's the base two model...

base two pieces

Of course, the pattern can extend in both directions. Notice that we've used exponents to illustrate the fact that each piece's value can be determined by multiplying '2' together a number of times. For example, 64 = 2x2x2x2x2x2 and 16 = 2x2x2x2, and so forth.

What quantities do these base 2 numbers represent?

Try to represent these quantities with base 2 models and notation:

The use of exponents is fundamental to place value. In base 8, for example, the digit '6' can mean various things, depending on where it is placed in the number. To avoid confusion, we organize our number systems into columns so that each column has a particular value, that is, it represents a certain piece in our collection of base 'n' pieces. The following table shows how this works...

(The columns represent the value of each location in a number)

base two
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
20 = 1

43 =
1 (1x32)
0
1 (1x8)
0
1 (1x2)
1 (1x1)

base four
45 = 1,024
44 = 256
43 = 64
42 = 16
41 = 4
40 = 1

2750 =
2 (2x1024)
2 (2x256)
2 (2x64)
3 (3x16)
3 (3x4)
2 (2x1)

base seven
75 = 16,807
74 = 2,401
73 = 343
72 = 49
71 = 7
70 = 1

55,555 =
3 (3x16,807)
2 (2x2,401)
0
6 (6x49)
5 (5x7)
3 (3x1)

base ten
105 = 100,000
104 = 10,000
103 = 1,000
102 = 100
101 = 10
100 = 1

273,964 =
2 (2x100,000)
7 (7x10,000)
3 (3x1,000)
9 (9x100)
6 (6x10)
4 (4x1)

In the base seven number, for example, the digit '3' represents a different value, depending upon which column it occupies. The first '3' represents 3 x 16,807 units, and the second '3' represents just 3 units.

In base notation form the numbers would be written

The procedure would be the same for any base we choose. The values of the columns are equal to the base number raised to a certain power. The exponent in each column is always the same, no matter what the base is.