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Binary Numbers

Binary, hexadecimal, octal, and decimal representation of numbers

Positive Numbers

Decimal to binary conversion of positive numbers

Let's convert the decimal number 99 to binary:
99₁₀ = 1*64 + 1*32 + 0*16 + 0*8 + 0*4 + 1*2 + 1*1 = 1*2⁶ + 1*2⁵ + 0*2⁴ + 0*2³ + 0*2² + 1*2¹ + 1*2⁰.
or 99₁₀ = 1100011₂
Decimal to binary conversion of positive numbers practice

Binary to decimal conversion of positive numbers

Let's convert the binary unsigned number 1100110₂ from binary to decimal.
Any unsigned binary number b_nb_n-1...b₁b₀ can be converted to decimal with the following formula:

b_nb_n-1...b₁b₀ = b_n×2ⁿ + b_n-1×2^n-1 + ... + b₀×2⁰,

1100110₂ = 1×2⁶ + 1×2⁵ + 0×2⁴ + 0*2³ + 1*2² + 1×2¹ + 0*2⁰ = 64 + 32 + 4 + 2 = 102.
Binary to decimal conversion of positive numbers practice

Hexadecimal to binary conversion of positive numbers

Convert an integer number from hexadecimal to binary by simply translating each hexadecimal digit into its 4-bit binary equivalent. Hexadecimal numbers have either an 0x prefix, a 16 subscript or an h suffix.
For example, the hexadecimal number 0x9E3 translates into 1001 1110 0011, as the binary values of 9, A and 3 are 1001, 1110 and 0011.
Hexadecimal to binary conversion of positive numbers

Binary to hexadecimal conversion of positive numbers

To convert a value from binary to hexadecimal, you first add a number of 0's to the left of the most significant bit of the binary number, so the number of bits of the new binary number is a multiple of 4. Then, you simply translate each 4-bit binary nibble to its hexadecimal digit equivalent.
Binary to hexadecimal conversion of positive numbers test

Hexadecimal to decimal conversion of positive numbers

A hexadecimal number can be converted to decimal by following the method:
0xh_nh_n-1...h₁h₀ where h_i is any of 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F , can be converted to the decimal number:
h_n×16ⁿ + h_n-1×16^n-1 + ... + h₁×16 + h₀.
Hexadecimal to decimal conversion of positive numbers

Decimal to hexadecimal conversion of positive numbers

The simplest way to convert unsigned numbers from decimal to hexadecimal is to transform the number first in binary, as the binary to hex conversion is trivial.
For instance:
99₁₀ = 1*64 + 1*32 + 0*16 + 0*8 + 0*4 + 1*2 + 1*1 = 1*2⁶ + 1*2⁵ + 0*2⁴ + 0*2³ + 0*2² + 1*2¹ + 1*2⁰.
99₁₀ = 1100011₂ = 0110 0011₂ = 0x63.
Decimal to hexadecimal conversion of positive numbers

Signed Integers

Decimal to binary conversion of signed numbers

Let's convert to signed binary the following integer: -200
-200 = -256 + 32 +16 + 8 = = -1×2⁸ + 0×2⁷ + 0×2⁶ + 1×2⁵ + 1×2⁴ + 1×2³ + 0×2² + 0×2¹ + 0×2⁰ = 100111000₂.
Decimal to binary conversion of signed numbers practice

Binary to decimal conversion of signed numbers

Let's convert to decimal the following signed binary number: 10110010
10110010 = -1×2⁷ + 0×2⁶ + 1×2⁵ + 1×2⁴ + 0×2³ + 0×2² + 1×2¹ + 0×2⁰ = -128 + 32 + 16 + 2 = -78.
Binary to decimal conversion of signed numbers

Positive Fractions

Decimal to binary conversion of positive binary fractions

Let's convert to unsigned binary the following positive fraction : 0.6875.
.6875 = 1×0.5 + 1×0.125 + 1×0.0625 = 1×2^-1 + 0×2^-2 + 1×2^-3 + 1×2^-4 = 0.1011
Decimal to binary conversion of positive binary fractions

Binary to decimal conversion of positive binary fractions

Let's convert to decimal the following unsigned binary fraction: 0.1011
0.1011 = 1×2^-1 + 0×2^-2 + 1×2^-3 + 1×2^-4 = .5 + .125 + .0625 = .6875.
Binary to decimal conversion of positive binary fractions

Signed Binary Fractions

Decimal to binary conversion of signed binary fractions

Let's convert to binary the following negative fraction: -0.4375.
-0.4375 = -1 + .5625 = -1 + .5 + .0625 = -1 + 1×2^-1 + 0×2^-2 + 0×2^-3 + 1×2^-4 = 1.1001
Decimal to binary conversion of signed binary fractions

Binary to decimal conversion of signed binary fractions

Let's convert to decimal the following signed binary fraction: 1.01101.
1.01101 = -1 + 0×2^-1 + 1×2^-2 + 1×2^-3 + 0×2^-4 + 1×2^-5 = -0.59375.
Binary to decimal conversion of signed binary fractions

Operations with Binary Numbers

Unsigned Numbers Addition

Binary addition is done like adding decimal numbers, except that you have only two digits. You have to remember only that:
0+0 = 0, with no carry,
1+0 = 1, with no carry,
0+1 = 1, with no carry,
1+1 = 0, and you carry a 1.

Let's add the following unsigned binary numbers: 11111 and 10101.


Unsigned Numbers Addition

Signed Numbers Addition/Subtraction

Signed numbers in two's complement notation may be added by following these rules:
1. If the numbers have different signs, they are added as unsigned numbers, and no overflow will occur.
2. If the numbers have the same sign, they are added and the sign of the resulting number is the same as the sign of both numbers, no overflow occurred.
3. If the numbers have the same sign, they are added and the sign of the resulting number is different than the sign of both numbers, overflow occurred.
Signed Numbers Addition/Subtraction

Unsigned Numbers Multiplication

Multiplication can be done exactly as with decimal numbers, except that you have only two digits. The only facts to remember are that 0×1=0, and 1×1=1.
Let's multiply the following unsigned binary numbers: 101 and 10.

Unsigned Numbers Multiplication

Other Binary Math Resources

Introduction to Binary Numbers from the Carnegie Melon University.

Binary Numbers from CUNY.