Study this article if you need more practice at understanding number systems.
Familiarize the learner with method for expressing numbers and convert one method to another.
A writing method for expressing numbers is called a "numeral system". In the most common numeral system, we write numbers with combinations of 10 symbols {0,1,2,3,4,5,6,7,8,9}. These symbols are called digits, and numbers that are expressed using 10 digits are called "decimal" or "base-10" numbers. The other most common numeral systems are binary, hexadecimal, and octal. The binary numeral system, or base-2 number system, represents numeric values using two symbols: 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers.
In the first method discussed we write numbers with combinations of 10 symbols {0,1,2,3,4,5,6,7,8,9} called digits. Numbers that are expressed with 10 digits are called "base-10" numbers or "Decimal Numeral System". For example:
2 (one digit)
45 (two digit)
643 (three digit)
8785 (four digit)
etc.
In Decimal Numeral Systems, the value of a digit is multiplied according to its placement in a numerical sequence: (base-number ^ 0,1,2,3,...), from right to left.
First digit = (base-number ^ 0): 10^0 = 1
Second digit =(base-number ^ 1): 10^1 = 10
Third digit =(base-number ^ 2): 10^2 = 100
Fourth digit =(base-number ^ 3): 10^3 = 1000
etc.
For example:
20= (2*10)+(0*1)=20+0=20
456=(4*100)+(5*10)+(6*1)=400+50+6
84568=(8*10000)+(4*1000)+(5*100)+(6*10)+(8*1)=80000+4000+500+60+8
Numbers expressed with 2 symbols (0, 1) are called binary, or "base-2" numbers.
For example:
1 (one-digit-read: 1)
10 (two-digit-read: 1, 0)
100 (three-digit-read: 1,0,0)
1101 (four-digit-read: 1, 1, 0, 1)
etc.
In the Binary Numeral System, digits have a value specified, this value being equal with (base-number ^ 0,1,2,3,...): (right to left)
First digit (base-number^0): 2^0 = 1
Second digit (base-number^1): 2^1 = 2
Third digit (base-number^2): 2^2 = 4
Fourth digit (base-number^3): 2^3 = 8
etc.
To convert binary to decimal, each digit is multiplied by the value of its position, and the results are added.
For example:
10 = (1*2^1) + (0*2^0) = 1*2 + 0*1 = 2 + 0 = 2 → 10 (binary) =2 (decimal)
101 = (1*2^2) + (0*2^1) + (1*2^0) = 1*4 + 0*2 + 1*1 = 4 + 0 + 1 = 5 → 101 (binary) =5 (decimal )
11001 = (1*2^4) + (1*2^3) + (0*2^2) + (0*2^1) + (1*2^0) = 1*16 + 1*8 + 0*4 + 0*2 + 1*1 = 16 + 8 + 0 + 0 + 1 = 25 → 11001 (binary) =25 (decimal)
111011 = (1*2^5) + (1*2^4) + (1*2^3) + (0*2^2) + (1*2^1) + (1*2^0) = 1*32 + 1*16 + 1*8 + 0*4 + 1*2 + 1*1 = 32 + 16 + 8 + 0 + 2 + 1 = 59 → 111011 (binary) =59 (decimal)
To convert decimal to binary
Divide the decimal number by 2
Then repeat the process, moving one column to the left each time until you have divided down to 1.
Example 1
15/2 = 7 remainder 1 (Binary number = ???1)
7/2 = 3 remainder 1 (Binary number = ??11)
3/2 = 1 remainder 1 (Binary number = ?111)
The final result will always be 1 in the leftmost column (Binary number = 1111)
Example 2
74/2 = 37 remainder 0 (Binary number = ??????0)
37/2 = 18 remainder 1 (Binary number = ?????10)
18/2 = 9 remainder 0 (Binary number = ????010)
9/2 = 4 remainder 1 (Binary number = ???1010)
4/2
= 2 remainder 0 (Binary number = ??01010)
2/2 = 1 remainder 0 (Binary number = ?001010)
The final result will always be 1 in the leftmost column (Binary number = 1001010)
Example 1: The largest factor less than 74 is 64, which is 2 to the power 6. As the furthest right column is 2 to the power 0, this means we need 7 columns.
Numbers written with 16 symbols {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} are called "base-16" numbers. For example:
A (one digit)
B5 (two digit)
6C3 (three digit)
AF85 (four digit)
etc.
so:
A(hexadecimal)=10(decimal).
B(hexadecimal)=11(decimal).
C(hexadecimal)=12(decimal).
D(hexadecimal)=13(decimal).
E(hexadecimal)=14(decimal).
F(hexadecimal)=15(decimal)
In the "Hexadecimal Numeral System", digits have a value specified, this value of digits is equal with (base-number^ 0,1,2,3,...):(right to left)
First digit (base-number ^ 0): 16^0 = 1
Second digit (base-number ^ 1): 16^1 = 16
Third digit (base-number ^ 2): 16^2 = 256
fourth digit (base-number ^ 3): 16^3 = 4096
etc.
To convert hexadecimal to decimal, each digit is multiplied by the value of its position, and the results are added.
For example:
A = (10*16^0) = 10*1 = 10 → A(hexadecimal) =10(decimal)
B5 = (11*16^1) + (5*16^0) = 11*16 + 5*1 = 181 → B5(hexadecimal) =181(decimal)
6C3 = (6*16^2) + (12*16^1) + (3*16^0) = 6*256 + 12*16 + 3*1 = 1536 + 192 + 3 = 1731 → 6C3(hexadecimal) =1731(decimal)
AF85 = (10*16^3) + (15*16^2) + (8*16^1) + (5*16^0) = 10*4096 + 15*256 + 8*16 + 5*1 = 40960 + 3840 + 128 + 5 = 44933 → AF85(hexadecimal) =44933(decimal)
To convert decimal to hexadecimal
Examples
Source: Wikiversity, https://en.wikiversity.org/wiki/Numeral_systems
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