# مختلط اعداد کا کیلکولیٹر

> Add, subtract, multiply and divide complex numbers, raise them to powers, find nth roots, and convert a + bi to polar form, with an Argand diagram.

انٹرایکٹو صورت: https://www.calcopenly.com/ur/math/complex-number-calculator
موضوع: ریاضی کے کیلکولیٹر

A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. Addition, subtraction and multiplication work in rectangular form; division multiplies top and bottom by the conjugate of the divisor. Powers and roots use polar form, z = r(cos θ + i sin θ): De Moivre's formula raises the modulus r to the power n and multiplies the angle θ by n.

Electrical engineers use complex numbers for AC impedance and phasors, and algebra students meet them as roots of polynomials. The default multiplies (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i, whose modulus is √125 ≈ 11.1803.

Inputs can be rectangular (3 + 4i, with j accepted for i) or polar (2∠90). Angles follow the degree, radian or gradian setting, and the argument is given between −180° and 180°. The Argand diagram draws each number as an arrow from the origin.

## اندراجات

- **Operation** (اختیارات: Convert to polar form, Add z₁ + z₂, Subtract z₁ − z₂, Multiply z₁ × z₂, Divide z₁ ÷ z₂, Power z₁ⁿ, nth roots of z₁)
- **z₁**: Rectangular a + bi (j also works) or polar r∠θ; θ uses the angle unit from settings unless you add ° or rad
- **z₂**
- **n**: Power: any real number. Roots: a whole number from 1 to 360.

## نتائج

- نتیجہ — بنیادی نتیجہ
- Real part
- Imaginary part
- Modulus |z|
- Argument (angle)
- Polar form

## فارمولا

$$
\begin{gathered} z = a + bi = r(\cos\theta + i\sin\theta) \\[4pt] r = \sqrt{a^2+b^2},\quad \theta = \operatorname{atan2}(b, a) \\[10pt] z^n = r^n(\cos n\theta + i \sin n\theta) \end{gathered}
$$

## حل شدہ مثالیں

### (3 + 4i)(1 − 2i)

- Operation: Multiply z₁ × z₂
- z₁: 3 + 4i
- z₂: 1 - 2i
- **نتیجہ: 11 − 2i**
- **Real part: 11**
- **Imaginary part: -2**
- جانچ کا ماخذ: ⁨3 − 6i + 4i − 8i² = 11 − 2i (hand calculation; Python (3+4j)*(1-2j))⁩

### (3 + 4i) ÷ (1 − 2i)

- Operation: Divide z₁ ÷ z₂
- z₁: 3 + 4i
- z₂: 1 - 2i
- **نتیجہ: −1 + 2i**
- **Real part: -1**
- **Imaginary part: 2**
- جانچ کا ماخذ: ⁨Python (3+4j)/(1-2j) = (-1+2j)⁩

### Polar form of 3 + 4i

- Operation: Convert to polar form
- z₁: 3 + 4i
- **Modulus |z|: 5**
- **Argument (angle): 53.1301023542**
- جانچ کا ماخذ: ⁨Python decimal: atan2(4, 3) in degrees via Machin-series arctangent (hp.py)⁩

### (1 + i)⁸

- Operation: Power z₁ⁿ
- z₁: 1 + i
- n: 8
- **نتیجہ: 16**
- **Real part: 16**
- **Imaginary part: 0**
- جانچ کا ماخذ: ⁨Python (1+1j)**8 = (16+0j); (√2)⁸ = 16 at angle 8 × 45° = 360°⁩

### Cube roots of 8

- Operation: nth roots of z₁
- z₁: 8
- n: 3
- **نتیجہ: 2, −1 + 1.732050808i, −1 − 1.732050808i**
- **Real part: 2**
- **Imaginary part: 0**
- جانچ کا ماخذ: ⁨2·(cos 120k° + i sin 120k°): Python cmath.rect(2, 2πk/3)⁩

### Polar input 2∠90° (edge: exact zero real part)

- Operation: Convert to polar form
- z₁: 2∠90
- **Real part: 0**
- **Imaginary part: 2**
- **Modulus |z|: 2**
- **Argument (angle): 90**
- جانچ کا ماخذ: ⁨2(cos 90° + i sin 90°) = 2i⁩

## سوالات

### How do you multiply complex numbers?

Expand the brackets and replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (3 + 4i)(1 − 2i) that is (3 + 8) + (−6 + 4)i = 11 − 2i. In polar form the rule is shorter: multiply the moduli and add the angles, so 5∠53.13° × 2.236∠−63.43° = 11.18∠−10.30°.

### How do you divide complex numbers?

Multiply the top and bottom by the conjugate of the divisor, which makes the denominator a real number. (3 + 4i)/(1 − 2i) = (3 + 4i)(1 + 2i)/(1² + 2²) = (3 + 6i + 4i + 8i²)/5 = (−5 + 10i)/5 = −1 + 2i. Division by 0 + 0i is undefined.

### How do you convert a complex number to polar form?

The modulus is r = √(a² + b²) and the argument is θ = atan2(b, a), which picks the correct quadrant. For 3 + 4i, r = √(9 + 16) = 5 and θ ≈ 53.130°, so 3 + 4i = 5∠53.130°. Plain arctan(b/a) gives the wrong angle when the real part is negative: −3 − 4i has an argument of about −126.870°, not 53.130°.

### What is De Moivre's theorem?

For z = r(cos θ + i sin θ) and a whole number n, zⁿ = rⁿ(cos nθ + i sin nθ). So (1 + i)⁸, with r = √2 and θ = 45°, equals (√2)⁸ = 16 at an angle of 8 × 45° = 360°, which is the real number 16. The same idea gives n evenly spaced nth roots: the cube roots of 8 are 2 and −1 ± 1.732051i, 120° apart.

### What is the square root of −1?

The imaginary unit i, defined by i² = −1; −i is the other square root. Every negative number has two imaginary square roots, so √−4 = ±2i, and the principal root, the one with argument 90°, is 2i. Powers of i repeat every four steps: i, −1, −i, 1, then i again.

### “⁨مختلط اعداد کا کیلکولیٹر⁩” کتنا درست ہے؟

درستی آپ کی درج کردہ قدروں اور طریقے کے مفروضوں پر منحصر ہے۔ اعشاری حساب 50 بامعنی ہندسے استعمال کرتا ہے، مگر تخمینے، عددی طریقے اور ماخذ کا ڈیٹا کم درست ہو سکتے ہیں؛ دکھائی گئی قدروں کو راؤنڈ کرنے سے یہ حدود ختم نہیں ہوتیں۔ آزاد ذرائع کی حل شدہ مثالوں سے جانچ: 7۔ مثلاً، “⁨(3 + 4i)(1 − 2i)⁩” کو ⁨3 − 6i + 4i − 8i² = 11 − 2i (hand calculation; Python (3+4j)*(1-2j))⁩ سے جانچا جاتا ہے۔

### اس طریقے کا ماخذ کیا ہے؟

Wolfram MathWorld — Complex Number; Wolfram MathWorld — de Moivre's Identity.

## ماخذ

- [Wolfram MathWorld — Complex Number](https://mathworld.wolfram.com/ComplexNumber.html)
- [Wolfram MathWorld — de Moivre's Identity](https://mathworld.wolfram.com/deMoivresIdentity.html)