Pengantar Logika : Proposisi, Hubungan Antar Proposisi, Definisi Term, Sifat Term, dan Contoh Soal

Sistem Digital : Sistem Bilangan I ( Sistem Desimal, Konversi Bilangan Biner, Representasi Heksadesimal )

The Decimal System

System based on decimal digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent numbers. For example the number 83 means eight tens plus three :

83 = (8 * 10) + 3

The number 4728 means four thousands, seven hundreds, two tens, plus eight :

4728 = (4 * 1000) + (7 * 100) + (2 * 10) + 8

The decimal system is said to have a base, or radix, of 10. This means that each digit in the number is multiplied by 10 raised to a power corresponding to that digit’s position:

83 = (8 * 10^1) + (3 * 10^0)

4728 = (4 * 10^3) + (7 * 10^2) + (2 * 10^1) + (8 * 10^0)

Decimal Fractions

The same principle holds for decimal fractions, but negative powers of 10 are used. Thus, the decimal fraction 0.256 stands for 2 tenths plus 5 hundredths plus 6 thousandths :

0.256 = (2 * 10^-1) + (5 * 10^-2) + (6 * 10^-3) 

A number with both an integer and fractional part has digits raised to both positive and negative powers of 10 :

442.256 = (4 * 10^2) + (4 + 10^1) + (2 * 10^0) + (2 * 10^-1) + (5 * 10^-2) + (6 * 10^-3)

Most significant digit

The leftmost digit (carries the highest value)

Least significant digit

The rightmost digit

Positional Interpretation of a Decimal Number

Positional Number Systems

Each number is represented by a string of digits in which each digit position i has an associated weight ri, where r is the radix, or base, of the number system. The general form of a number in such a system with radix r is

( . . . a_3a_2a_1a_0.a_-1a_-2a_-3 . . . )r

where the value of any digit ai is an integer in the range 0 < ai < r. The dot between a_0 and a_-1 is called the radix point.

Positional Interpretation of a Number in Base 7

The Binary System

Only consists two digits, 1 and 0. Binary represented to the base of 2. The digits 1 and 0 in binary notation have the same meaning as in decimal notation :

0_2 = 0_10

1_2 = 1_10

To represent larger numbers each digit in a binary number has a value depending on its position :

10_2 = (1 * 2^1) + (0 * 2^0) = 210

11_2 = (1 * 2^1) + (1 * 2^0) = 310

100_2 = (1 * 2^2) + (0 * 2^1) + (0 * 2^0) = 410

and so on. Again, fractional values are represented with negative powers of the radix :

1001.101 = 2^3 + 2^0 + 2^-1 + 2^-3 = 9.62510

Converting Between Binary and Decimal

Binary Notation to Decimal notation :

Multiply each binary digit by the appropriate power of 2 and add the results

Decimal notation to binary notation :

Integer and fractional parts are handled separately

Integers

For the integer part, recall that in binary notation, an integer represented by

b_m-1b_m-2 . . . b_2b_1b_0

 b_i = 0 or 1

has the value

(b_m-1 * 2^m-1) + (b_m-2 * 2^m-2) + . . . + (b_1 * 2^1) + b_0

Suppose it is required to convert a decimal integer N into binary form. If we divide N by 2, in the decimal system, and obtain a quotient N_1 and a remainder R_0, we may write

N = 2 * N_1 + R_0 // R_0 = 0 or 1

Next, we divide the quotient N_1 by 2. Assume that the new quotient is N_2 and the new remainder R_1. Then :

N1 = 2 * N2 + R1 // R1 = 0 or 1

so that

N = 2(2N_2 + R_1) + R_0 = (N_2 * 2^2) + (R_1 * 2^1) + R_0

Because N >N_1 > N_2 . . . , continuing this sequence will eventually produce a quotient N_m-1 = 1 (except for the decimal integers 0 and 1, whose binary equivalents are 0 and 1, respectively) and a remainder R_m-2, which is 0 or 1. Then :

N = (1 * 2^m-1) + (R_m-2 * 2^m-2) + . . . + (R_2 * 2^2) + (R_1 * 2^1) + R_0

Which is the binary form of N. Hence, we convert from base 10 to base 2 by repeated divisions by 2. The remainders and the final quotient, 1, give us, in order of increasing significance, the binary digits of N.

Examples of Converting from Decimal Notation to Binary Notation for Integers

Fractions

Number with a value between 0 and 1 is represented by

0.b_-1b_-2b_-3 . . . b_i = 0 or 1

and has the value

(b_-1 * 2^-1) + (b_-2 * 2^-2) + (b_-3 * 2^-3) . . .

This can be rewritten as

2^-1 * (b_-1 + 2^-1 * (b_-2 + 2^-1 * ( b_-3 + . . . ) . . . )

Suppose we want to convert the number

F (0 < F < 1) from decimal to binary notation. We know that F can be expressed in the form

F = 2-1 * (b-1 + 2-1 * (b-2 + 2-1 * (b-3 + . . . ) . . . ))

If we multiply F by 2, we obtain,

2 * F = b_-1 + 2^-1 * (b_-2 + 2^-1 * (b_-3 + . . . ) . . . )

From this equation, we see that the integer part of (2 * F), which must be either 0 or 1 because 0 < F < 1, is simply b-1. So we can say (2 * F) = b-1 + F1, where 0 < F1 < 1 and where

F1 = 2-1 * (b-2 + 2-1 * (b-3 + 2-1 * (b-4 + . . . ) . . . ))

To find b−2, we repeat the process.

At each step, the fractional part of the number from the previous step is multiplied by 2. The digit to the left of the decimal point in the product will be 0 or 1 and contributes to the binary representation, starting with the most significant digit. The fractional part of the product is used as the multiplicand in the next step.

Examples of Converting from Decimal Notation to Binary Notation for Fractions

Hexadecimal Notation

Binary digits are grouped into sets of four bits, called a nibble. Each possible combination of four binary digits is given a symbol, as follows:

0000 = 0
0100 = 4
1000 = 8
1100 = C

0001 = 1
0101 = 5
1001 = 9
1101 = D

0010 = 2
0110 = 6
1010 = A
1110 = E

0011 = 3
0111 = 7
1011 = B
1111 = F


Because 16 symbols are used, the notation is called hexadecimal and the 16 symbols are the hexadecimal digits. Thus,

2C_16 = (2_16 * 16^1) + (C_16 * 16^0) = (2_10 * 16^1) + (12_10 * 16^0) = 44

Decimal, Binary, and Hexadecimal Table

The Reason on using Hexadecimal Notation

Sumber

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Slide AOK : Sistem Bilangan