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Computer Organization Cheatsheet

Definitions, Symbols, Formulas, and Notes — All in One Place.

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Binary

Term		Definition		Example
Binary Number		A number expressed in the base-2 numeral system, which uses only two numbers, 0 and 1.
Most Signficant Bit MSB		The leftmost bit in a binary number.
Least Significant Bit LSB		The rightmost bit in a binary number.
Binary to Decimal Conversion		For a binary number $n$ , use the formula $n = \sum_{i = 0}^{k} a_{i} \times 2^{i}$ , where $a_{i}$ is each octal digit and $i$ is its position in the number (where the rightmost digit is position 0).		Convert $1100101101_{2}$ to hexadecimal. Grouping into sets of four: $0011 0010 1101$ Convert each group to decimal: $0011_{2} = 3_{16}$ $0010_{2} = 2_{16}$ $1101_{2} = D_{16}$ $1100101101_{2} = 32 D_{16}$
Binary to Octal Conversion		Start from the rightmost binary digit and group them into sets of three. If there are not enough digits in the leftmost group, pad with zeros. Convert each group to decimal and map to each corresponding octal digit.		Convert $1011101_{2}$ to octal. Grouping into sets of three: $001 011 101$ Convert each group to its octal equivalent: $001_{2} = 1_{8}$ $011_{2} = 3_{8}$ $101_{2} = 5_{8}$ So, $1011101_{2} = 135_{8}$ in octal.
Binary to Hexadecimal Conversion		Start from the rightmost binary digit and group them into sets of four. If there are not enough digits in the leftmost group, pad with zeros. Convert each group of 4 to decimal and then map the result to the corresponding hexadecimal digit.		Convert $1100101101_{2}$ to hexadecimal. Grouping into sets of four: $0011 0010 1101$ Convert each group to decimal: $0011_{2} = 3_{16}$ $0010_{2} = 2_{16}$ $1101_{2} = D_{16}$ $1100101101_{2} = 32 D_{16}$

Octal

Term		Definition		Examples
Octal		A base-8 numeral system. It uses the digits 0 through 7 to represent numbers.
		Octal to Binary Conversion		Begin by converting the octal number into its decimal form, then follow the rules for converting from decimal to binary.		Convert $127_{8}$ to binary. 1. Convert each octal digit to its 3-digit binary representation: $1_{8} = 001_{2}$ $2_{8} = 010_{2}$ $7_{8} = 111_{2}$ $127_{8} = 001010111_{2}$
Octal to Hexadecimal Conversion		Begin by converting the octal number into its decimal form, then follow the rules for converting from decimal to hexadecimal.		Convert $543_{8}$ to hexadecimal. Convert $5_{8}$ to binary: $5_{8} = 101_{2}$ Convert $4_{8}$ to binary: $4_{8} = 100_{2}$ Convert $3_{8}$ to binary: $3_{8} = 011_{2}$ Combine: $543_{8} = 101100011_{2}$ Group into fours: $1011 0001 1$ Convert each group to hexadecimal. $1011_{2} = B_{16}$ $0001_{2} = 1_{16}$ $1_{16} = 1_{16}$ $543_{8} = B 11_{16}$
Octal to Decimal Conversion		For an octal number $n$ , use the formula $n = \sum_{i = 0}^{k} a_{i} \times 8^{i}$ , where $a_{i}$ is each octal digit and $i$ is its position in the number (where the position of the rightmost digit is 0).		Convert $647_{8}$ to decimal. $(6 \times 8^{2}) + (4 \times 8^{1}) + (7 \times 8^{0})$ $= (6 \times 64) + (4 \times 8) + (7 \times 1)$ $= 384 + 32 + 7$ $= 423$ $647_{8} = 423_{10}$

Decimal

Term		Definition		Examples
Decimal		A number expressed in a base 10 numeral system which uses digits 1-9. We usually use decimal numbers in our day-to-day lives.
Decimal to Octal Conversion		Divide the original number by largest power of 8 possible, then divide the resulting remainder by a smaller power of 8. Repeat this process until the largest power possible is 0, taking each successive quotient as one octal digit, starting from the left.		Convert $315_{10}$ to octal. $315_{10} \div 8 = 39 (quotient) with 3 (remainder)$ $39 \div 8 = 4 (quotient) with 7 (remainder)$ $4 \div 8 = 0 (quotient) with 4 (remainder)$ $315_{10} = 473_{8}$
Decimal to Hexadecimal Conversion		Repeatedly divide the decimal number by 16, noting the remainders as hexadecimal digits from right to left, until the quotient is zero.		Convert $789_{10}$ to hexadecimal. $789_{10} \div 16 = 49 (quotient) with 5 (remainder)$ $49 \div 16 = 3 (quotient) with 1 (remainder)$ $3 \div 16 = 0 (quotient) with 3 (remainder)$ $789_{10} = 315_{16}$
Decimal to Binary Conversion		Continuously divide by 2 until arriving at a quotient of 1. The remainder after each division (including the final quotient) represents one binary digit, starting from the left.		Convert $92_{10}$ to binary. $92_{10} \div 2 = 46 (quotient) with 0 (remainder)$ $46 \div 2 = 23 (quotient) with 0 (remainder)$ $23 \div 2 = 11 (quotient) with 1 (remainder)$ $11 \div 2 = 5 (quotient) with 1 (remainder)$ $5 \div 2 = 2 (quotient) with 1 (remainder)$ $2 \div 2 = 1 (quotient) with 0 (remainder)$ $1 \div 2 = 0 (quotient) with 1 (remainder)$ $92_{10} = 1011100_{2}$ in binary.

Hexadecimal

Term		Definition		Examples
Hexadecimal		Hexadecimal is a base-16 numeral system. It uses the digits 0 through 9 and the letters A through F (corresponding to 10-16) to represent numbers.
Hexadecimal to Binary Conversion		Substitute each hex digit with its 4-digit binary representation.		$3_{16} = 0011_{2}$ $F_{16} = 1111_{2}$ $3 F_{16} = 00111111_{2}$
Hexadecimal to Octal Conversion		Convert the hexadecimal number to binary, then group the binary digits into groups of three starting from the right. Then, map each group of three to its octal equivalent.		First, convert $A_{16}$ to binary: $A_{16} = 1010_{2}$ Then, convert $5_{16}$ to binary: $5_{16} = 0101_{2}$ Combine: $A 5_{16} = 10100101_{2}$ Group into threes: $101 001$ Convert each group to octal: $101_{2} = 5_{8}$ $001_{2} = 1_{8}$ $A 5_{16} = 151_{8}$
Hexadecimal to Decimal Conversion		For an octal number $n$ , use the formula $n = \sum_{i = 0}^{k} a_{i} \times 8^{i}$ , where $a_{i}$ is each octal digit and $i$ is its position in the number (where the position of the rightmost digit is 0).		Convert $B 2_{16}$ to decimal. $B 2_{16} = (11 \times 16^{1}) + (2 \times 16^{0})$ $= (11 \times 16) + (2 \times 1)$ $= 176 + 2$ $= 178$

Floating Point

Boolean Algebra

Term		Definition		Formula
De Morgan's Law		The first law states that the negation of the logical OR of two propositions is equivalent to the logical AND of their negations. The second law states that the negation of the logical AND of two propositions is equivalent to the logical OR of their negations.		$\neg (A \lor B) = \neg A \land \neg B$ $\neg (A \land B) = \neg A \lor \neg B$
Karnaugh Map K-Map		A graphical representation of a truth table used to simplify Boolean algebraic expressions.
Sum of Products SOP		Expresses a logic function as the logical OR (sum) of multiple logical AND (products) terms.
Product of Sums POS		Expresses a logic function as the logical AND (product) of multiple logical OR (sums) terms.
Identity Law		States that a Boolean variable ANDed with $1$ is equal to the variable itself, and a Boolean variable ORed with $0$ is equal to the variable itself.		$A \lor 0 = A$ $A \land 1 = A$
Indempotent law		States that a Boolean variable ORed with itself is equal to the variable itself, and a Boolean variable ANDed with itself is equal to the variable itself.		$A \lor A = A$ $A \land A = A$
Domination Law		States that a Boolean variable ORed with true $1$ is equal to true, and a Boolean variable ANDed with $0$ is equal to false.		$A \lor 1 = 1$ $A \land 0 = 0$
Null Law		States that		$A \land 0 = 0$ $A \lor 1 = 1$
Commutative Law		States that a Boolean variable ORed with itself is equal to the variable itself, and a Boolean variable ANDed with itself is equal to the variable itself.		$A \land B = B \land A$ $A \lor B = B \lor A$
Associative Law		States that grouping of Boolean variables in a series of OR or AND operations does not affect the result.		$(A \land B) \land C = A \land (B \land C)$ $(A \lor B) \lor C = A \lor (B \lor C)$
Distributive Law		Describes how to expand expressions involving both OR and AND operations by distributing one operation over the other.		$A \land (B \lor C) = (A \land B) \lor (A \land C)$ $A \lor (B \land C) = (A \lor B) \land (A \lor C)$
Absorption Law		States that combining a Boolean variable with its conjunction or disjunction with another variable results in the original variable.		$A \land (A \lor B) = A$ $A \lor (A \land B) = A$

Logic Gates

Signed Integers

Term		Definition
One's Complement		A method for representing signed binary numbers where the negative of a binary number is obtained by flipping all the bits in its binary representation.
Two's Complement		A method for representing signed binary numbers where the negative of a binary number is obtained by taking the one's complement of the number and then adding 1 to the result. This representation has the advantage that there is only one representation for zero and that addition and subtraction can be performed using the same hardware.
Excess Notation		A method for representing signed integers where a bias value is added to the unsigned representation of a number to obtain its signed representation.
Signed Magnitude		A method for representing signed integers where the MSB is used to represent the sign. A value of $0$ typically denotes a positive number and $1$ denotes a negative number.

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