The Complex Conjugate Calculator provided by Hesapstan finds z̄=a−bi for z=a+bi, then shows z × z̄ and the modulus squared |z|²=a²+b². It does not compute the actual modulus |z|=√(a²+b²), polar form, general complex arithmetic, or a complex-plane plot.
The complex conjugate changes only the sign of the imaginary part
The complex conjugate of a number a+bi keeps the real part a unchanged and reverses the sign of the imaginary part. Therefore, if z=a+bi, then z̄=a−bi.
Here a is the real part, while b is the coefficient of the imaginary unit i. Taking the conjugate does not change a; it changes +bi to −bi, or −bi to +bi.
The conjugate of a+bi is a−bi: keep the real part, flip the sign of the imaginary part.
This calculator shows z̄, z × z̄, and |z|²
The tool asks for two numeric inputs: a, the real part, and b, the imaginary coefficient. It then displays the original z, the conjugate z̄, the product z × z̄, and the same value labeled as the modulus squared |z|².
- Enter the real part a.
- Enter the imaginary coefficient b; negative and decimal values are allowed.
- Read z=a+bi and its conjugate z̄=a−bi.
- Use the z × z̄ row to see why the product becomes a real number.
- Do not read the |z|² row as |z|; it is the squared modulus, not the square-rooted modulus.
The modulus |z| is √(a²+b²). This calculator shows only |z|²=a²+b², so the displayed value is not the actual length unless you take the square root separately.
The product z × z̄ is always real
Multiplying a complex number by its conjugate cancels the imaginary terms. Expanding (a+bi)(a−bi) gives −abi and +abi, which cancel. Since i²=−1, the remaining expression is a²+b².
That is why z × z̄ is both a real number and equal to the modulus squared |z|². This relationship appears in rationalizing complex denominators, interpreting conjugate roots, and norm-like calculations.
For z=3+4i, the conjugate is z̄=3−4i. The product is (3+4i)(3−4i)=3²+4²=9+16=25. This 25 is |z|²; the actual modulus |z| is 5, but this calculator displays the squared value.
Special cases make the conjugate rule easier to check
The conjugate rule is the same for real, purely imaginary, and zero values, but the output becomes simpler. These are useful checks when reading the result.
- If b=0, then z=a is real, and its conjugate is the same number.
- If a=0, then z=bi is purely imaginary, and its conjugate is −bi.
- If a=0 and b=0, then z=0, z̄=0, z × z̄=0, and |z|²=0.
- If b is negative, the sign still flips: the conjugate of 2−5i is 2+5i.
Modulus squared is not the same as modulus
The modulus squared is |z|²=a²+b². The modulus itself is |z|=√(a²+b²). According to the calculator scope, this page displays the squared value, not the square root.
This distinction matters because z × z̄ directly gives a²+b². Until a square root is taken, the value is a squared length, not the actual length in the complex plane.
If the result says |z|²=25, that does not mean the modulus is 25. The modulus would be 5, but this calculator's official output is the squared modulus.
Complex roots of real-coefficient polynomials come in conjugate pairs
For a polynomial with real coefficients, a non-real complex root is accompanied by its conjugate. In a quadratic equation with negative discriminant, the two roots usually appear as p+qi and p−qi.
When the discriminant D is negative, a quadratic has no real roots; it has two complex conjugate roots instead. This calculator does not solve those roots, but it explains the conjugate structure behind them.
This is not a general complex-number calculator
The scope is intentionally narrow: from one complex number z, the calculator returns z̄, z × z̄, and |z|². It does not add, multiply, divide, graph, or convert complex numbers to polar form.
- No polar form r and θ are calculated.
- No general operations such as z₁+z₂, z₁×z₂, or z₁/z₂ are performed.
- Symbolic a and b coefficients are not supported; the inputs must be numeric.
- No complex-plane graph or angle output is included.
Frequently Asked Questions
What is the complex conjugate of a+bi?
The complex conjugate of a+bi is a−bi. The real part stays the same, and the imaginary part changes sign.
Does this calculator compute the modulus |z|?
No. It computes the modulus squared |z|²=a²+b². The modulus |z|=√(a²+b²) is not displayed as a separate output.
Why is z × z̄ always real?
Because the imaginary terms cancel when (a+bi)(a−bi) is expanded, leaving a²+b².
Is z × z̄ the same as |z|²?
Yes. For every complex number z, z × z̄ = |z|² = a²+b².
What is the conjugate of a real number?
If b=0, the number is real and its conjugate is itself. For example, the conjugate of 7 is 7.
What is the conjugate of a purely imaginary number?
If a=0, then z=bi and the conjugate is −bi. For example, the conjugate of 5i is −5i.
Can this calculator divide complex numbers?
No. General complex arithmetic such as addition, multiplication, or division is outside this calculator's scope.
Why do complex roots appear as conjugates?
Non-real roots of real-coefficient polynomials occur in conjugate pairs, which is why negative-discriminant quadratics have roots of the form p±qi.