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Sign-Magnitude Converter

Convert signed decimal numbers to Sign-Magnitude binary format and decode sign-magnitude bits. Designed for software developers, electrical engineers, students, and computer architecture researchers requiring deterministic client-side accuracy.

Engine: Client-Side Verified (0ms Latency)

Enter a signed decimal integer or binary string.

Calculated Output

Primary Representation
Calculating...

Step-by-Step Mathematical Proof

Active Derivation
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Concept

What Is Sign-Magnitude Converter?

Sign-Magnitude (or signed magnitude) representation dedicates the most significant bit (MSB) as a sign flag (0 for positive, 1 for negative) while the remaining bits encode the absolute magnitude of the number in standard binary.

Methodology

How Does It Work?

For an n-bit register: determine the sign (0 or 1). Convert the absolute magnitude |N| into an (n-1)-bit binary string. Combine the sign bit with the magnitude string.

Formula & Rules

Mathematical Algorithm

X = (-1)^s \times \sum_{i=0}^{n-2} b_i 2^i \quad \text{where } s = b_{n-1}
Worked Problem

Step-by-Step Example

Convert -27 to 8-bit sign-magnitude: 1) Sign is negative -> MSB = 1 2) Magnitude |27| in 7-bit binary: 0011011 3) Combine: 1 0011011 -> 10011011.

Important Rules & Edge Cases

  • MSB is strictly the sign bit (0 = positive, 1 = negative).
  • Remaining n-1 bits represent magnitude.
  • Has dual zeros: +0 is 00000000 and -0 is 10000000.

Practical Applications in Engineering

  • Mantissa representation in IEEE-754 floating-point numbers.
  • Analog-to-digital converter (ADC) signed output codes.
  • Audio sample sign encoding in early digital signal processors.

Common Mistakes to Avoid

  • Caution: Confusing sign-magnitude with Two’s Complement (in sign-magnitude, the magnitude bits are NOT inverted).
  • Caution: Overflowing the (n-1) magnitude bit limit.
FAQ

Frequently Asked Questions

Why was sign-magnitude abandoned for integer arithmetic in CPUs?

Because it requires separate circuits for addition and subtraction, and has two zeros (+0 and -0), which complicates zero-testing logic.

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