Boolean Algebra

Logic gates and circuits The "AND" gate, the "OR" gate and the "NOT" gate
Binary number system Binary to decimal conversion Decimal to binary conversion
Logic Gates and Circuits
There are exactly three basic electronic circuits called logic gates each of which correspond to one of the three Boolean (binary) operators, “and,” “or,” and “not” having the same properties.
AND gate
 a b a · b 0 0 0 1 0 0 0 1 0 1 1 1
OR gate
 a b a + b 0 0 0 1 0 1 0 1 1 1 1 1
NOT gate (or invertor)   a a' 0 1 1 0
Logic circuits used in digital computers are built up from logic gates. We want to know the output y of a logic circuit for all possible combinations of input bits. The value of the output is shown at the resultant column of the corresponding truth table.
Binary Number System
The binary number system uses digits, 0 and 1 to represent numbers. A binary number can be therefore represented by any sequence of bits (binary digits).
The tables show binary representations of integers from 0 to 19 with corresponding place values of bits.
 Decimal number Binary number 24 23 22 21 20 0 0 0 0 0 0 1 0 0 0 0 1 2 0 0 0 1 0 3 0 0 0 1 1 4 0 0 1 0 0 5 0 0 1 0 1 6 0 0 1 1 0 7 0 0 1 1 1 8 0 1 0 0 0 9 0 1 0 0 1
 Decimal number Binary number 24 23 22 21 20 10 0 1 0 1 0 11 0 1 0 1 1 12 0 1 1 0 0 13 0 1 1 0 1 14 0 1 1 1 0 15 0 1 1 1 1 16 1 0 0 0 0 17 1 0 0 0 1 18 1 0 0 1 0 19 1 0 0 1 1
Binary to decimal conversion:  Any binary number can be converted to its decimal equivalent by writing it in a place-value notation, i.e. as the sum of products of each digit with place value of that digit.
 Example: = 1 · 26 +  0 · 25 + 1 · 24 + 1 · 23 + 1 · 22 +  0 · 21 + 1 · 20 =
=    64   +     0    +   16    +    8    +    4    +     0    +     1    = 93
 Decimal to binary conversion: To convert a decimal number to its binary equivalent divide given decimal and each successive quotient by 2 noting remainders from right to left, that is from the lowest place value to the higher. The remainders can only be 0 and 1 since divisions are by 2. The division ends by the quotient zero.
 113 ÷ 2 = 1 1 1 1 0 0 0 1 56 <==== 28 14 7 3 1 0   Beginning Algebra Contents 