Numbers in Bases Two, Eight and Five (Part 1)

(A) Numbers in Bases Two, Eight and Five
1. The numbers we use daily are in base ten. The ten digits used in numbers 
     in based ten are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
2. Numbers in base two are numbers that use two digits only. The digits are 0 
    and 1. Example: 1011012.
3. Numbers in base eight are numbers that use eight digits only. The eight 
    digits used in numbers in based ten are 0, 1, 2, 3, 4, 5, 6, and 7. 
    Example: 6758.
4. Numbers in base five are numbers that use five digits only. The five digits 
    used in numbers in based five are 0, 1, 2, 3, and 4. Example: 1345.


(B) Value of a Digit of a Number in Base 2, 8 and 5
1. The following tables show the place values of the digits of a number
 (a) In base 2
Place value
26 = 64
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
20 = 1

(b) In base 8
Place value
84 = 4096
83 = 512
82 = 64
81 = 8
80 = 1

(c) In base 5
Place value
54 = 625
53 = 125
52 = 25
51 = 5
50 = 1
2.
 Value of a digit = The digit × Place value of the digit
Example 1:
State the value of the underlined digit in each of the following numbers.
(a) 10111012                   (b) 36518               (c) 32415

Solution:
 (a)

Place value
26 = 64
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
20 = 1
Number
1
0
1
1
1
0
1
= 1 × 26The value of the underlined digit 1
= 1 × 64
= 64

(b)


Place value
83 = 512
82 = 64
81 = 8
80 = 1
Number
3
6
5
1
= 6 × 82
The value of the underlined digit 6
= 6 × 64
= 384

(c)

Place value
53 = 125
52 = 25
51 = 5
50 = 1
Number
3
2
4
1
= 3 × 53
The value of the underlined digit 3
= 3 × 125
= 375


(C) Writing a Number in Base 2, 8, or 5 in Expanded Notation
1. A number written in expanded notation refers to the sum of the value of 
    the digits that make up the number.

Example 2:
Write each of the following in expanded notation.
(a) 1110112                     (b) 4758                (c) 24135

Solution:
(a)

Place value
25 = 32
24 = 16
23 = 8
22 = 4
21 = 2
20 = 1
Number
1
1
1
0
1
1
1110111 × 2+ 1 × 2+ 1 × 2+ 0 × 2+ 1 × 2+ 1 × 20

(b)


Place value
82 = 64
81 = 8
80 = 1
Number
4
7
5
4758 = 4 × 8+ 7 × 8+ 5 × 80

(c) 

Place value
53 = 125
52 = 25
51 = 5
50 = 1
Number
2
4
1
3
24132 × 5+ 4 × 5+ 1 × 5+ 3 × 50

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