Compute sgn(x) for any number or expression, see absolute value connection, product-of-signs rule, power rule, number line visualization, sign table for a range, and properties reference.
The **Sign Function Calculator** evaluates the signum function sgn(x), which extracts the sign of any real number: +1 for positive numbers, −1 for negative numbers, and 0 for zero. Despite its simplicity, the sign function is a fundamental building block in analysis, signal processing, optimization algorithms, and computer graphics, wherever you need to separate a value's magnitude from its direction.
This calculator goes beyond a simple sign check. It supports four expression modes: evaluate the sign of a single number, a product, a difference, or a quotient. It displays the absolute value alongside the sign, verifies the decomposition identity x = sgn(x) × |x|, computes the product of signs for two numbers, confirms the multiplicative property sgn(ab) = sgn(a) × sgn(b), and evaluates sgn(xⁿ) for any integer power.
A color-coded number line shows where your value sits relative to zero, with red for the negative half-line and green for the positive. The range table steps through consecutive values and displays sign, absolute value, and the identity product with colored indicator dots. The properties reference table collects the key identities: decomposition, multiplicativity, idempotence, the odd-function property, and the power rule.
Whether you are implementing a comparator in code, analyzing the convergence direction of a sequence, or teaching students the concept of positive and negative, this tool gives you a complete, visual exploration of the sign function and its properties.
This calculator is useful because the sign function is simple in definition but widely used in proofs, algorithms, and piecewise modeling. The calculator does not stop at a single positive-or-negative label. It can evaluate a single number, a product, a difference, or a quotient, which makes it a practical way to inspect how sign changes under common operations.
It is also strong as a teaching tool. The decomposition identity, product-of-signs check, power rule, number line, and range table all show how sign is separated from magnitude. That helps when explaining absolute value, direction, comparator logic, and the behavior of expressions near zero.
sgn(x) = +1 if x > 0, −1 if x < 0, 0 if x = 0. Decomposition: x = sgn(x) × |x|. Multiplicative: sgn(xy) = sgn(x) × sgn(y).
Result: sgn(42) = 1 and |42| = 42.
Because 42 is positive, the sign function returns +1. The identity x = sgn(x) × |x| becomes 42 = 1 × 42.
The sign function strips away magnitude and keeps only direction relative to zero. A positive input maps to +1, a negative input maps to -1, and zero maps to 0. This is useful whenever you care about orientation, direction, or comparison rather than size.
Because of that, the sign function appears in analysis, numerical methods, optimization, and software that needs to branch on whether a value is above, below, or equal to zero.
One of the most important identities is $x = operatorname{sgn}(x) imes |x|$. It shows that any real number can be reconstructed from its sign and its magnitude. This calculator displays that identity directly, which helps explain why sign and absolute value are naturally paired concepts.
The number line view reinforces the same idea visually by showing where the input sits relative to zero and which side of the axis determines the sign.
The sign of a product depends on the signs of its factors, and the sign of a power depends on both the base and whether the exponent changes the sign pattern. The expression modes for products, differences, and quotients let you inspect those cases directly, while the dedicated power output shows how $operatorname{sgn}(a^n)$ behaves for your chosen input.
The range table then extends the idea across many values, making it easier to see where sign flips occur and how zero separates negative and positive regions.
The sign function (sgn) returns −1 for negative numbers, 0 for zero, and +1 for positive numbers. It extracts the sign of a real number: sgn(x) = x / |x| for x ≠ 0, so it separates direction from magnitude.
sgn(0) = 0 by convention. Zero is neither positive nor negative, so the sign function returns 0 as a special case that preserves the three-valued rule.
It appears in direction logic, piecewise formulas, optimization routines, signal processing, and physics models where only orientation matters. Any time you need a compact positive/negative/zero test, the sign function is useful.
The signum function returns −1 for negative inputs, 0 for zero, and +1 for positive inputs. This three-valued result is useful whenever the direction of a number matters more than its magnitude.
The sign function is used in sorting algorithms to determine ordering, in physics simulations to set force direction, and in optimization routines to compute subgradients and update step directions. It is especially useful when only direction matters and magnitude can be handled separately.
No, the sign function is discontinuous at zero where it jumps from the limit of −1 (approaching from the left) to +1 (approaching from the right). It is therefore not differentiable at zero.