Question

Write the complex number in polar form. $$\sqrt{3}+i$$

   Write the complex number in polar form.
$$\sqrt{3}+i$$

Precalculus

Jay Abramson 1st Edition

Chapter 8, Problem 326

Instant Answer

Solved by Expert Mohamed Mohamed

07/21/2019

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In this case, $x=\sqrt{3}$ and $y=1$. Show more…

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Write the complex number in polar form. $$\sqrt{3}+i$$

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Key Concepts

Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, typically expressed in the form a + bi. They extend the real number system and allow for solutions to equations that involve the square roots of negative numbers.

Polar Form of a Complex Number

The polar form represents a complex number using a magnitude (or modulus) and an angle (or argument). Instead of expressing the number in terms of its rectangular coordinates (a, b), it is expressed in terms of its distance from the origin and the angle it makes with the positive real axis.

Modulus

The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It is calculated using the Pythagorean theorem as the square root of the sum of the squares of the real and imaginary parts.

Argument

The argument of a complex number is the angle formed between the positive real axis and the line that represents the complex number. This angle, usually measured in radians, can be determined using inverse trigonometric functions, such as arctan, based on the ratio of the imaginary part to the real part.

Conversion from Rectangular to Polar Form

Converting a complex number from rectangular form (a + bi) to polar form involves calculating the modulus and the argument. The modulus is found using the formula |z| = sqrt(a^2 + b^2) and the argument is determined using inverse trigonometric functions. The number can then be represented in the form r(cos ? + i sin ?) or r·e^(i?), which is advantageous for operations like multiplication and division.

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