Number & Operations for Teachers Copyright David & Cynthia Thomas, 2009 |
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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?
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.
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
585 > 111 Þ 1111 eight > 111
292 > 222 Þ 444 eight > 222
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. |
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