Number & Operations for Teachers 

    Copyright David & Cynthia Thomas, 2009

Counting in Base Eight

 

Imagine that on some other distant planet, intelligent spiders count using their eight legs as follows. 

Zero, one, two, three, four, five, six, seven.

Spidy, Spidy one, Spidy two, Spidy three, ... and so on.

 

Fill in Table 1.5 using the same sort of thinking used when developing base five counting (See Table 1.3).

-          How is their counting language similar to base five?  How is it different?

-          How is their counting notation similar to base five?  How is it different?

-          How would you write the number that comes immediately after 77eight? 

-          What would you name it?

 

0eight

Zero

10eight

Spidy

 

 

 

 

 

 

1eight

One

11eight

Spidy one

 

 

 

 

 

 

2eight

Two

12eight

Spidy two

 

 

 

 

 

 

3eight

Three

13eight

Spidy three

 

 

 

 

 

 

4eight

Four

 

 

 

 

 

 

 

 

5eight

Five

 

 

 

 

 

 

 

 

6eight

Six

 

 

 

 

 

 

 

 

7eight

Seven

 

 

 

 

 

 

 

 

Table 1.5: Counting in Base Eight

 

Complete Table 1.6 by showing the value of each numeral in the number 4321eight.  The value associated with each column is determined by an eight (written 10eight) raised to some integral power.

4

3

5

1

4000 eight

300 eight

50 eight

1 eight

4(1000 eight)

3(100 eight)

5(10 eight)

1(1 eight)

4(10 eight 3)

3(10 eight 2)

5(10 eight 1)

1(10 eight 0)

Table 1.6: Representation for 4321eight

 

Example 1.3

Compare the following quantities and indicate which is larger:

a.       1111eight  and 111

b.      444 eight  and 222

c.       2000 eight and 1000

 

Solution 1.3

a.       1111 eight  =

1(83) + 1(82) + 1(81) + 1(80) =

1(512) + 1(64) + 1(8) + 1(1) = 585

111 =

1(102) + 1(101) + 1(100) =

1(100) + 1(10) + 1(1) = 111

585 > 111 Þ

1111 eight  > 111

 

b.      444 eight  =

4(82) + 4(81) + 4(80) =

4(64) + 4(8) + 4(1) = 292

222 =

2(102) + 2(101) + 2(100) =

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

292 > 222 Þ

444 eight  > 222

 

c.       2000 eight  =

2(83) + 0(82) + 0(81) + 0(80) =

2(512) + 0(64) + 0(8) + 0(1) = 1024

1000 =

1(103) + 0(102) + 0(101) + 0(100) =

1(1000) + 0(100) + 0(10) + 0(1) = 1000

1024 > 1000 Þ

2000 eight  > 1000

 

 

 

Example 1.4

Convert 121 to base eight

 

Solution 1.4

This conversion, which involves division, may be thought of as repeated subtraction.  Begin by identifying the largest power of eight that divides into 121, in this case 64.  This division yields a quotient of 1 and a remainder of 57.  The remainder is then divided by 8, the next smaller power of 8, yielding a quotient of 7 and a remainder of 1.  The next small power of 8, one, is then divided into this remainder.  This sequence of quotients shows that 121 = 1(82) + 7(81) + 1(80) = 171five.  This procedure is summarized below: 

 

On the basis of these examples, it should now be clear that there is nothing inherently special about base ten numeration.  Indeed, it seems likely that this system arose simply because humans have ten fingers on their hands, i.e., as a matter of convenience.  As to why the ancient Sumerians developed a base sixty system, we may never know.