# GNED 1101 Chapter Notes - Chapter 2.2: 5,6,7,8, Hexadecimal

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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