 # 1.1 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