Number system is made up of different digit(called symbol) that are applicable for computation within a system. THE HIGHEST DIGIT OF ANY NUMBER SYSTEM IS ONE LESS THAN THE BASE.
The following are the basic number system
*BINARY SYSTEM*(Base 2) : symbol used are 0 and 1. It has place values that are power of 2
*Octal system"(Base 8): Symbol used are 0,1,2,3,4,5,6 and 7. It has a place value which has the power of 8
*Denary system* (Base 10): Symbol used are 0,1,2,3,4,5,6,7,8 and 9. It has a place value which has the power of 10
*Duodecimal system*(Base 12):. It has a place value which has the power of 12. Symbol used are 0,1,2,3,4,5,6,7,8,9,X,and A
Where X = 10 and A = 11
*Hexadecimal system*(Base 16):. It has place values which has the power of 16.
Symbol used are: 0,1,2,3,4,5,6,7,8,9,A, B,C,D,E,F
Where, A=10, B=11, C=12, D =13, E=14, F=15
*Conversion from other bases to base 10*
RULE:
*worked examples*:
convert the following to base 10
(1)32145(base six)
Solution:
By expansion,we have:
3×6⁴+2×6³+1×6²+4×6¹+5×6°
=3888+432+36+24+5
=4385(base 10)
Hope you understand!
2.) 100101(base2)
By expansion,
1×2⁵+0×2⁴+0×2³+1×
2²+0×2¹+1×2°
=32+0+0+4+0+1
=37(base 10)
B· *conversion from base 10 to other bases*
Example 3:
Convert 89₁₀ to a binary number (base 2)
Solution:
By long division,we didvide by the new base(which is 2)
2√89
2÷44 r (1)
2÷22 r (0)
2÷11 r (0)
2÷5 r (1) ∆
2÷2 r (1) |
2 ÷1 r (0) |
0 r (1)
Following the arrow's direction we have:1011001₂(base2)
Example 4:
Convert 75₁₀ to an octal number(base 8)
Solution:
Divide by the new base:
75÷8=9 r 3
9 ÷8=1 r 1
1 ÷8=0 r 1
=113₈(base8)
C. Conversion from a number which is not in base ten to another number ( not denary)
*WHEN CONVERTING A NUMBER WHICH IS NOT IN BASE 10 TO A BASE WHICH IS NOT DENARY. YOU FIRST CONVERT TO BASE 10, THEN TO THE REQUIRED BASE.
Example 5:
Express 307₈(base8)
in binary.
Solution:
We're to convert from base 8 to base 2
Step 1: convert to base 10 by expansion.
= 3×8²+0×8¹+7×8°
=192+0+7
=199(base 10)
Step 2: convert 199₁₀(base10)
to base 2 by long division.
Solution:
199÷2=99 r 1
99÷ 2=49 r 1
49÷ 2=24 r 1
24÷2= 12 r 0
12÷2=. 6 r 0
6÷ 2= 3 r 0
3÷ 2= 1 r 1
1 ÷2= 0 r 1
= 11000111₂(base2)
Example 6
Express 111111(base2) in octal.:
Solution:
Convert to base 10
=1×2^5+1×2⁴+1×2³+1×2²+1×2¹+1×2°
=32+16+8+4+2+1
=63(base10)
Then convert 63 to base 8 by dividing by 8
= 63÷8=7 r 7
7÷8=0 r 7
= 77(base 8)
Example 7:
Convert 14312(base5) to base 6
Solution:
Step 1:first convert it to base 10.
= 1×5⁴+4×5³+3×5²+
1×5¹+2×5°
=625+500+75+5+2
=1207(base10)
Step 2:Then convert 1207 to base 6 by continuous division by 6
=
1207÷6=201 r1
201÷6=33 r3
33÷6=5 r3
5÷6=0 r 5
=5331(base 6)
(D)BI - DEXIMALS:
These are decimals in binary form (base 2)
Note-When converting any number to number in base 10 that include decimal fraction,we multiply each digit of the number by the corresponding power of the base it is expressed.
8. Convert 0.234₅ to base 10
Solution-
0.234₅ = 0 x 5⁰ + 2 x 5⁻¹ + 3 x 5⁻² + 4 x 5⁻³
= 0 + 2/5 + 3/25 + 4/125
= 69/125₁₀ or 0.552₁₀
Example 9:
Solution:
Convert 1001.11(base two) to base 10
=1×2³+1×2²+1×2¹+
1×2°+1×2^-¹ +1×2^-²
(The powers for the numbers after the decimal point are negative because they are less than 1, i.e,1001.11=1001+0.11)
=8+0+0+1+½+1/4
=9+¾=9 ¾
Example 10
Convert 4et(base12) to base 10
Solution:
[e stands for 11 and
t stands for 10]
= 4×12²+e×12¹+t×12°
=4×144+11×12+10×1
=718(base 10)
E. *Addition, subtraction, multiplication and division in number bases*.
Example 11:
Simplify: 136+425+243{all in base 7}
Solutions:
Step 1:convert all to base 10
={1×7²+3×7¹+6×7°}
+{4×7²+2×7¹+5×7°}
+{2×7²+4×7¹+3×7°}
=49+21+6+196+14+5+98+28+3
=420{base ten}
Step 2:convert 420 to base7 by division.
= 420÷7=60 r 0
60÷7=8. r 4
8÷7=1 r 1
1÷7=0 r 1
=1140₇(base7)
Example 12:
Simplify: 324-135₆{base 6}
Solutions:
Step 1:convert all to
base 10 by expansion
={3×6²+2×6¹+4×6°}
-{1×6²+3×6¹+5×6°}
={124}- {59}
=65₁₀ (base ten)
Step 2:convert 65₁₀ to base 6
=
65÷6=10 r 5
10÷6= 1. r 4
1÷6= 0 r 1
= 145₆
Example 13:
Find the product of 235₆and 14₆
Solution:
Convert all to base ten
235six=2×6²+3×6¹+5×6°=95ten
14₆ =1×6¹+4×6°
=10₁₀
=95×10=950₁₀
Step 2:convert 950₁₀ to base six.
=950÷6=158 r 2
158÷6=26. r 2
26÷6= 4 r 2
4÷6= 0. r 4
=4222₆
Example 14
Divide 10010001₂ by 101two.
Solution:
convert both to base ten and divide.Then change the result to base 2
By expansion,10010001two=
145ten and
101two=5ten.
= 145÷5=29ten
= 29÷2=14 r1
14÷2=7 r0
7÷2 =3 r1
3÷2= 1 r1
1÷2= 0 r1
=11101two
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