GNED 1101 Chapter Notes - Chapter 2.2: 5,6,7,8, Hexadecimal
8 Jun 2018
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2.2 Number Bases in Positional Systems
Changing Numerals in Bases Other then Ten to Base Ten
• The base of a positional numeration system refers to the number of individual digit
symbols that can be used in that system as well as to the number whose powers define
the place value
o Example, the number 1001two
▪ Is not read as one thousand one because that is of the base 10 system.
▪ Going forward, when a numeral does not have a subscript attached to it
it is assumed to be of the base ten.
▪ However. Since the two is subscripted to the base two, the number is
read as one zero zero one base two.
• In any base system, the digit symbols begin at 0 and go up to one less than the base.
Table 4 (p.125)
Base
2Digit Symbols
Place Values
Two
0, 1
…, 4, 23, 22, 21, 1
Three
0, 1, 2
…, 4, 33, 32, 31, 1
Four
0, 1, 2, 3
…,4, 43, 42, 41, 1
Five
0, 1, 2, 3, 4
…,4, 53, 52, 51, 1
Six
0, 1, 2, 3, 4, 5
…, 4, 63, 62, 61, 1
Seven
0, 1, 2, 3, 4, 5, 6
…, 4, 73, 72, 71, 1
Eight
0, 1, 2, 3, 4, 5, 6, 7
…, 4, 83, 82, 81, 1
Nine
0, 1, 2, 3, 4, 5, 6, 7, 8
…, 4, 93, 92, 91, 1
Ten
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
…, 4, 103, 102, 101, 1
• Follow these steps to change a base to base ten
o Find the place value for each digit in the numeral
o Multiply each digit in the numeral by its respective place value
o Find the sum of the products in step 2
• Examples;
o Convert 4726eight to base 10
▪ The numeral has 4 place values. From left to right the place values are 83,
82, 81, and 1
▪ Multiply each digit in the numeral by its respective place value, then find
the sum of these products
▪ 4726eight = (4 x 83) + (7 x 82) + (2 x 81) + (6 x 1)
= (4 x 8 x 8 x8) + (7 x 8 x 8) + (2 x 8) + (6 x 1)
= (2048) + 448 + 16 + 6
= 2518
o Convert 100101two
o 100101two. = (1 x 25) + (0 x 24) + (0 x 23) + (1 x 22) + (0 x 21) + (1 x 1)
= (1 x 32) + (0 x 16) + (0 x8) + (1 x 4) + (0 x 2) + ( 1 x 1)
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