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Binary Representation Basics
Understanding Binary Numbers
Binary System:
The binary number system is base-2, using only digits 0 and 1. Each digit is referred to as a bit.
Converting Decimal to Binary:
To convert a decimal number to binary, divide the number by 2 and record the remainder. Repeat with the quotient until it is zero, then read the remainders in reverse.
// Decimal to Binary Example
int decimal = 10;
String binary = Integer.toBinaryString(decimal);
System.out.println(binary); // Output: 1010
Console Output:
1010
Binary Arithmetic Operations
Addition:
Binary addition follows the same rules as decimal addition but is simpler as it involves only two digits. The possible sums are 0+0=0, 0+1=1, 1+0=1, and 1+1=10 (which is 0 with a carry of 1).
// Binary Addition Example
int a = 0b1010; // 10 in decimal
int b = 0b1100; // 12 in decimal
int sum = a + b;
System.out.println(Integer.toBinaryString(sum)); // Output: 10110
Console Output:
10110
Binary Subtraction
Subtraction:
Binary subtraction uses borrowing, similar to decimal subtraction. If the top bit is smaller than the bottom bit, borrow 1 from the next left bit.
// Binary Subtraction Example
int a = 0b1100; // 12 in decimal
int b = 0b1010; // 10 in decimal
int difference = a - b;
System.out.println(Integer.toBinaryString(difference)); // Output: 10
Console Output:
10
Binary Multiplication
Multiplication:
Binary multiplication is similar to decimal multiplication. Multiply each bit of one number by each bit of the other, then add the results, shifting them according to their position.
// Binary Multiplication Example
int a = 0b101; // 5 in decimal
int b = 0b11; // 3 in decimal
int product = a * b;
System.out.println(Integer.toBinaryString(product)); // Output: 1111
Console Output:
1111
Binary Division
Division:
Binary division is similar to long division in decimal. It involves dividing the dividend by the divisor, subtracting the result, and bringing down the next bit.
// Binary Division Example
int dividend = 0b1100; // 12 in decimal
int divisor = 0b11; // 3 in decimal
int quotient = dividend / divisor;
System.out.println(Integer.toBinaryString(quotient)); // Output: 100
Console Output:
100
