(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)
= 1 × 26The value of the underlined digit 1
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 × 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
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 |
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 |
(b)
Place value |
82 = 64 |
81 = 8 |
80 = 1 |
Number |
4 |
7 |
5 |
(c)
Place value |
53 = 125 |
52 = 25 |
51 = 5 |
50 = 1 |
Number |
2 |
4 |
1 |
3 |