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### Binary

To encode numbers in binary format there are a set of rules that must be fulfilled:

Decimal to Binary, e.g. 47:
47 ÷ 2 = 23, remainder = 1
23 ÷ 2 = 11, remainder = 1
11 ÷ 2 = 5, remainder = 1
5 ÷ 2 = 2, remainder = 1
2 ÷ 2 = 1, remainder = 0
1

Wrap up the remainders from bottom up, 47 in decimal is equal to 101111 in binary notation.

Binary to Decimal, e.g. 101111:
1 × 2^5 + 0 × 2^4 + 1 × 2^3 + 1 × 2^2 + 1 × 2^1 + 1 = 47

Decimal Fraction to Binary Fraction, e.g. 0.6875:
0.6875 * 2 = 1.375, non-fractional part = 1 //remove one
0.375 * 2 = 0.75, non-fractional part = 0
0.75 * 2 = 1.5, non-fractional part = 1
0.5 * 2 = 1.0, non-fractional part = 1

Weap up remainders from top to bottom: 0.1011

Binary Fraction to Decimal Fraction, e.g. 0.1011:
1 * 2^(-1) + 0 * 2^(-2) + 1 * 2^(-3) + 1 * 2^(-4) = 0.6875

Negative Using 8 bits
 Positive 0000,1001  //to show number 9 Signed magnitude 1000,1001  //just set the bit sign in the left most bit Signed 1's complement 1111,0110   //invert all bits, This scheme has two representations for 0 (0 and -0) Signed 2's complement 1111,0111   //start from right, leave all zeros until  you face 1, leave that one, invert all the rest to the left