Digital Principles: Number Systems

• The following notes cover digital principles.

Number Systems

• The binary number system is a weighted system that uses two digits.
• The octal number system consists of eight digits, 0–7.
• The 1's complement of the binary number 1010 is 0101.
• The hexadecimal number system has 16 characters, with six of them being alphabetic.
• A decimal number can be converted to binary using repeated division by 2.
• The decimal equivalent of the binary number 1101 is 13.
• Decimal number 21 is equivalent to 10101 in binary.
• The binary equivalent of hexadecimal 1F is 11111.
• The sum of the binary numbers 1101 + 0110 is 10011.
• The number system with a base of 8 is the Octal system.
• The hexadecimal equivalent of decimal 255 is FF.
• 57 in decimal is 111001 in binary.
• 3A in hexadecimal is 58 in decimal.
• 110011 in binary is 33 in hexadecimal.
• 765 in octal is 111110101 in binary.
• 101011 in binary is 53 in octal.
• A9 in hexadecimal is 10101001 in binary.
• 1010 + 1101 equals 10111 in binary.
• 1001 - 0111 equals 0010 in binary.
• 110 * 101 equals 100110 in binary.
• 00100001 + 10111100 equals 11011101 in binary.
• 01110111 - 00110010 equals 01000101 in binary.
• A NOR gate produces a high output only when all inputs are low.
• The logic gates in a circuit are defined using Boolean expressions.

Logic Gates

• An AND gate outputs 1 if at least one input is 1 (FALSE).
• An XOR gate produces an output of 1 when both inputs are 1 (FALSE).
• The output of an AND gate is always the same as the input when there is only one input.
• A NOR gate produces an output of 1 when both inputs are 0.
• A NAND gate outputs 0 when both inputs are 1.
• A NAND gate can implement any Boolean function.
• An AND gate produces a high output only when all inputs are high.
• For an AND gate with three inputs, the output is only 1 when all three inputs are 1.
• In an OR gate with three inputs, the output is 1 when at least one input is 1.
• The output of an XOR gate with inputs A = 1 and B = 1 is 0.
• The output of a NAND gate with inputs A = 0 and B = 1 is 1.
• An OR gate has the same truth table as a NOR gate followed by an inverter.
• The Boolean expression A + AB simplifies to A.
• If an AND gate has three inputs and one of them is 0, the output is 0.
• If an OR gate produces a 0 output when both inputs are 0, the gate functions correctly (TRUE).
• A logic circuit that produces an output of 1 when an odd number of inputs are 1 uses an XOR gate.
• A faulty NAND gate that always outputs 0 typically has one or both inputs stuck at 1.
• If an inverter (NOT gate) has an input of 1, the output is 0.
• A circuit needing an output of 1 when both inputs are different requires an XOR gate.
• DeMorgan's theorem applied to (A + B + C)' results in A'B'C'.
• The logic circuit for the expression AB + A'B is an XOR gate.
• The simplified form of the expression AB + A(B + C) is AB + AC.
• A NAND gate can be converted to an AND gate by using an inverter (NOT gate) after the NAND gate.
• The output is 0 in a truth table for an XOR gate when both inputs are 0.

Boolean Algebra

• Boolean algebra analyzes and simplifies digital logic circuits.
• The distributive law: A(B + C) = AB + AC.
• DeMorgan's theorem: (AB)' = A' + B'. Simplified Boolean Algebra:
• The commutative law states that A + B = B + A.
• DeMorgan's theorem applied to (A + B)' results in A'B'.
• The absorption law dictates that A(A + B) = A.
• A + AB simplifies to A.
• (AB)' is equivalent to A' + B' based on DeMorgan's theorem.
• The expanded form of the Boolean equation A(B + C) is AB + AC.
• Using a Karnaugh map, A B C + ABC + ABC + ABC simplifies to A + BC.
• For the function X = A + B, X is 1 if either A or B is 1.
• For the function X = AB, X is 1 if both A and B are 1.
• The binary output of a NAND gate with inputs A = 1 and B = 1 is 0.
• The binary output of a NOR gate with inputs A = 0 and B = 0 is 1.
• A 3-input AND gate's output is 1 only when all inputs are 1.
• A 3-input OR gate's output is 1 when at least one input is 1.
• If A = 1, B = 0, and C = 1, the output of X = AB + C is 1.
• A NOT gate inverts the input.
• The function X = A'B equals the truth table {A=0, B=1, X=1}.