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Category Math & Algebra

Operand A (Binary)

0 bits

Operand B (Binary)

0 bits

Presets:

Step-by-Step Breakdown

NOR Result (Binary)

NOR Result (Decimal)

NOR Result (Hexadecimal)

NOR Result (Octal)

Intermediate Values

A (padded)

B (padded)

A OR B

NOT(A OR B) = NOR

Truth Table (per bit)

Bit #	A	B	A OR B	NOR

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About

The NOR gate is a universal logic gate. Any Boolean function can be constructed using only NOR operations, making it foundational in digital circuit design. The NOR of two bits A and B equals NOT(A OR B). It outputs 1 only when both inputs are 0. Miscalculating a single NOR bit in a hardware description language propagates errors through entire adder or multiplexer chains. This tool operates on binary strings up to 64 bits, automatically zero-pads operands to equal length, and returns both the binary result and its decimal equivalent.

Note: this calculator assumes unsigned integer representation. Two's complement signed interpretation is not applied. For inputs exceeding 64 bits, truncation occurs at the most significant end. Pro Tip: verify NOR results against AND/OR/NOT decompositions when debugging multi-gate circuits - a single inverted bit in a NOR cascade can flip an entire bus line.

Formulas

The bitwise NOR operation processes two binary operands bit by bit. For each bit position i, the result bit R_i is computed as follows:

R_i = ¬(A_i ∨ B_i)

This is equivalent to the following piecewise definition:

{

R_i = 1 if A_i = 0 ∧ B_i = 0R_i = 0 otherwise

The full result is the concatenation of all R_i from the most significant bit to the least significant bit. Operands are zero-padded on the left to match the length of the longer operand before computation. Decimal conversion uses positional notation:

D = n−1∑i=0 b_i ⋅ 2ⁱ

Where A_i is the i-th bit of operand A, B_i is the i-th bit of operand B, R_i is the i-th bit of the result, n is the total number of bits after padding, b_i is the bit value at position i (from LSB), and D is the unsigned decimal equivalent.

Reference Data

Gate	Symbol	Expression	Output when A=0,B=0	Output when A=0,B=1	Output when A=1,B=0	Output when A=1,B=1	Universal?
AND	∧	A ⋅ B	0	0	0	1	No
OR	∨	A + B	0	1	1	1	No
NOT	¬	¬A	1	Unary operator			No
NAND	¬∧	¬(A ⋅ B)	1	1	1	0	Yes
NOR	¬∨	¬(A + B)	1	0	0	0	Yes
XOR	⊕	A ⊕ B	0	1	1	0	No
XNOR	¬⊕	¬(A ⊕ B)	1	0	0	1	No
Buffer	→	A	0	Passes input			No
NOR (3-input)	¬∨	¬(A+B+C)	1 only when all inputs are 0				Yes
NOT via NOR	¬	A NOR A	1	0	0	0	-
OR via NOR	∨	(A NOR B) NOR (A NOR B)	0	1	1	1	-
AND via NOR	∧	(A NOR A) NOR (B NOR B)	0	0	0	1	-
CMOS NOR	-	2 PMOS series + 2 NMOS parallel	Transistor count: 4				-
TTL 7402	-	Quad 2-input NOR	Supply: 4.75 - 5.25 V				-
ECL NOR	-	Emitter-coupled logic	Propagation delay: <1 ns				-

Frequently Asked Questions

NOR is defined as NOT(OR). The OR gate outputs 0 only when both A and B are 0. Inverting that single 0 yields 1. All other input combinations produce OR = 1, which inverts to 0. This makes NOR the most restrictive two-input gate.

When operand A has fewer bits than B, leading zeros are prepended to A. Since 0 NOR x equals NOT(x), those padded positions effectively invert the corresponding bits of the longer operand. Failing to pad correctly will produce a result with an incorrect bit width and wrong decimal value.

Yes. NOR is functionally complete (a universal gate). NOT is achieved by wiring the same signal to both NOR inputs: A NOR A = ¬A. OR is constructed by applying NOR twice: (A NOR B) NOR (A NOR B). AND requires three NOR gates: (A NOR A) NOR (B NOR B). Any combinational or sequential circuit can be built from NOR gates alone, as proven by Charles Sanders Peirce in 1880.

This tool supports up to 64 bits per operand. JavaScript's native number type uses IEEE 754 double-precision, which safely represents integers up to 2⁵³ − 1. For values beyond that, the tool uses BigInt for decimal conversion, ensuring exact results across the full 64-bit range without floating-point truncation.

NAND outputs 0 only when all inputs are 1; NOR outputs 1 only when all inputs are 0. In cascaded operations, NOR tends to produce results biased toward 0 bits because any 1 in either operand forces that bit position to 0. NAND shows the opposite bias. This distinction matters in carry-lookahead adder design where gate choice affects propagation delay and fan-out requirements.

The operation is applied independently to each corresponding bit position after alignment. Bit order (MSB-first vs LSB-first) does not affect the result as long as both operands use the same convention. This tool uses MSB-first (standard binary notation). If you are working with LSB-first data (common in some serial protocols), reverse your input strings before entering them.