Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Counting in Base Five

Imagine that on some distant planet, intelligent beings have only one hand. On each hand there are five fingers. And they use a numeration system based on five numerals. These people teach their children to count as follows:

Zero, one, two, three, four.

Handy, handy one, handy two, handy three, handy four.

Two handy, two handy one, two handy two, two handy three, two handy four.

Three handy, three handy one, three handy two, three handy three, three handy four.

Examine Table 1.3 in search of consistent patterns.

· How is their counting language similar to our own? How is it different?

· How is their counting notation similar to our own? How is it different?

· How would you write the number that comes immediately after 44 _five?

· What would you name it?

0five	Zero	10five	One Hand	20five	Two hand	30five	Three hand
1five	One	11five	Handy one	21five	Two handy one	31five	Three handy one
2five	Two	12five	Handy two	22five	Two handy two	32five	Three handy two
3five	Three	13five	Handy three	23five	Two handy three	33five	Three handy three
4five	Four	14five	Handy four	24five	Two handy four	34five	Three handy four

Table 1.3: Counting in Base Five

Numbers written without identifying subscripts are assumed to be expressed in base ten. For numbers in any other system, the system base is written as a subscript. For example,

· In base ten, the number following 9 is written 10

· In base five, the number following 4 _five is written 10_five

· In base ten, the number following 99 is written 100

· In base five, the number following 44 _five is written 100 _five

The structure of the base five numeration system is similar to that for the base ten numeration system in that the value associated with each column is determined by a five (written 10_five) raised to some integral power (See Table 1.4).

4	3	5	1
4000 five	300 five	50 five	1 five
4(1000 five)	3(100 five)	5(10 five)	1(1 five)
4(10 five^3)	3(10 five^2)	5(10 five^1)	1(10 five^0)

Table 1.4: Base Ten Representations for 4321_five

This pattern applies in all positional numeration systems. For instance, the first three columns in each of the following numeration systems represent the first three integral powers of the system’s base. To facilitate comparison, all of the numbers are written in base ten form.

· Base Ten: 10² 10¹ 10⁰

· Base Five: 5² 5¹ 5⁰

· Base Two: 2² 2¹ 2⁰

A natural question is, “How does the base of a numeration system affect the representations of identical and different quantities? “ For instance, because the numerals in the base five number 4,321_five are identical to the numerals in the base ten number 4,321 one might be tempted to think that they represent identical quantities. This idea can be tested by converting both numbers to the same base and comparing their magnitudes. In this case, the number 4,321_five may be converted to base ten as follows:

4(5³) + 3(5²) + 2(5¹) + 1(5⁰) =

4(125) + 3(25) + 2(5) + 1(1) = 586.

So, the number 4,321_five is considerably less than the base ten number 4,321.

How can the same numerals represent different quantities in different numeration systems? Consider the following metaphor. A child has two sets of blocks, one red and the other blue. Both sets have the same number of blocks, but the blocks in the red set are half the size of the blocks in the blue set. While the child could build identically shaped red and blue buildings, the blue building would be taller than the red building because the blue blocks are bigger. The same principle applies when building numbers, so 4,321_five < 4,321.

Example 1.1 demonstrates a procedure for converting a base five number to an equivalent base ten number. Example 1.2 demonstrates a related process for converting a base ten number to an equivalent base five number. In these examples, the symbol Þ should be read “therefore.”

Example 1.1

Compare the following quantities and indicate which is larger:

a. 1111_five and 111

b. 444_five and 222

c. 2000_five and 1000

Solution 1.1

a. 1111five = 1(5^3) + 1(5^2) + 1(5^1) + 1(5^0) = 1(125) + 1(25) + 1(5) + 1(1) = 156

111 = 1(10^2) + 1(10^1) + 1(10^0) = 1(100) + 1(10) + 1(1) = 111

156 > 111 Þ

1111_five > 111

b. 444five = 4(5^2) + 4(5^1) + 4(5^0) = 4(25) + 4(5) + 4(1) = 126

222 = 2(10^2) + 2(10^1) + 2(10^0) = 2(100) + 2(10) + 2(1) = 222

126 < 222 Þ

444_five < 222

a. 2000five = 2(5^3) + 0(5^2) + 0(5^1) + 0(5^0) = 2(125) + 0(25) + 0(5) + 0(1) = 250

1000 = 1(10^3) + 0(10^2) + 0(10^1) + 0(10^0) = 1(1000) + 0(100) + 0(10) + 0(1) = 1000

250 < 1000 Þ

2000_five < 1000

Example 1.2

Convert the following base ten number to base five

- 121

Solution 1.2

- Begin by identifying the largest power of five that divides into 121. Since 5³ is greater than 121 and 5² is less than 121, the largest power of five that divides into 121 is 5², or 25. In fact, there are 4 twenty-fives in 121. Subtracting this amount from 121 leaves a balance of 21. We then determine how many 5’s are in 21 by dividing 5 into 21. Using this process, we obtain 121 = 4(5²) + 4(5¹) + 1(5⁰) = 441_five.