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Complex number

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Algebra and Trigonometry

Definition

A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with property $i^2 = -1$. Complex numbers can be represented in both rectangular and polar forms.

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5 Must Know Facts For Your Next Test

  1. The modulus of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$.
  2. The polar form of a complex number is expressed as $r(\cos \theta + i\sin \theta)$ or $re^{i\theta}$, where $r = |z|$ and $\theta$ is the argument of the complex number.
  3. Addition and subtraction of complex numbers follow the rules $(a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i$.
  4. Multiplication of two complex numbers $(a+bi)(c+di)$ results in $(ac-bd) + (ad+bc)i$.
  5. The conjugate of a complex number $z = a + bi$ is given by $\overline{z} = a - bi$, useful for dividing complex numbers.

Review Questions

  • What is the modulus of the complex number $3 + 4i$?
  • Convert the complex number $1 - i$ to its polar form.
  • How do you find the product of two complex numbers?
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