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)
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.
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
|
1110112 = 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20
(b)
Place
value
|
82 = 64
|
81 = 8
|
80 = 1
|
Number
|
4
|
7
|
5
|
4758 = 4 × 82 + 7 × 81 + 5 × 80
(c)
Place
value
|
53 = 125
|
52 = 25
|
51 = 5
|
50 = 1
|
Number
|
2
|
4
|
1
|
3
|
24135 = 2 × 53 + 4 × 52 + 1 × 51 + 3 × 50
(D) Converting Numbers in Base Two, Eight and Five to Base Ten and Vice Versa
1. Steps to convert numbers in base 2, 8 and 5 to base 10 are as follows.
(a) write the number in expanded notation.
(b) simplify the expanded notation into a single number.
Example 1:
Convert each of the following numbers to a number in base 10.
(a) 101012 (b) 14238 (c) 3245
Solution:
(a) 101012 = 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 2110
(b) 14238 = 1 × 83 + 4 × 82 + 2 × 81 + 3 × 80 = 78710
(c) 3245 = 3 × 52 + 2 × 51 + 4 × 50 = 8910
2. Steps to convert a number in base 10 to a number in base 2, 8 and 5 are as fo
llows.
(a) perform repeated division until the quotient is zero.
(b) write the number in new base by referring to the remainders from bottom to the top.
Example 2:
Convert 6110 to a number in
(a) Base two (b) base eight (c) base five
Solution:
(a)
(b)
(c)
Calculator Computation
1. Set the calculator to the ‘BASE’ mode by pressing:
[MODE] [MODE] [3 (BASE)]
2. Set the calculator to the desired number system by pressing:
[BIN] → for base 2
[DEC] → for base 10
[OCT] → for base 8
Key in the following:
(a)
[DEC] 61 [=] [ BIN ]
The screen display is: [1111012]
Therefore 6110 = 1111012
(b)
[DEC] 61 [=] [ OCT ]
The screen display is: [75]
Therefore 6110 = 758
|
(E) Converting from One Base to Another
1. The following steps are used to convert a number from one base to another base.
(a) convert the number to a number in base 10 by using expended notation.
(b) use repeated division to convert the number in base 10 to the respective bases.
Example 1:
Convert
(a) 1101012 to a number in base 5
(b) 435 to a number in base 2
(c) 3138 to a number in base 5
(d) 4225 to a number in base 8
(e) 1001112 to a number in base 8
(f) 1578 to a number in base 2
(f) 1578 to a number in base 2
Solution:
(a) 1101012
= 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20
= 5310 ← (Convert from base 2 to base 10)
(b) 435
= 4 × 51 + 3 × 50
= 2310 ← (Convert from base 5 to base 10)
(c) 3138
= 3 × 82 + 1 × 81 + 3 × 80
= 20310 ← (Convert from base 8 to base 10)
(d) 4225
= 4 × 52 + 2 × 51 + 2 × 50
= 11210 ← (Convert from base 5 to base 10)
(e) 1001112
= 1 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 20
= 3910 ← (Convert from base 2 to base 10)
(f) 1578
= 1 × 82 + 5 × 81 + 7 × 80
= 11110 ← (Convert from base 8 to base 10)
credit : http://spmmathematics.onlinetuition.com.my/2015/01/numbers-in-bases-two-eight-and five.html
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