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Magazine Chapter 17: Problem 9
In Problems 1-10, write the given complex number in polar form. $$ \frac{3}{-1+i} $$
Short Answer
Expert verified
The polar form is \(\frac{3\sqrt{2}}{2} \text{cis} \frac{5\pi}{4}\).
Step by step solution
01
Multiply to remove the imaginary number from the denominator
To simplify the division by a complex number, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(-1+i\) is \(-1-i\). So the expression becomes:\[\frac{3}{-1+i} \times \frac{-1-i}{-1-i} = \frac{3(-1-i)}{(-1+i)(-1-i)}\]
02
Simplify the denominator
The denominator is a difference of squares, so:\[(-1+i)(-1-i) = (-1)^2 - (i)^2 = 1 - (-1) = 2\]
03
Simplify the numerator
Multiply out the numerator:\[3(-1-i) = -3 - 3i\]
04
Write the simplified form of the complex number
Combine the simplified numerator and denominator:\[-\frac{3 + 3i}{2} = -\frac{3}{2} - \frac{3}{2}i\]
05
Express the complex number in polar form
First, calculate the magnitude (modulus) of the complex number \(-\frac{3}{2} - \frac{3}{2}i\):\[|z| = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(-\frac{3}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{18}{4}} = \frac{3\sqrt{2}}{2}\]Next, calculate the argument:Since the number is in the third quadrant,\[\theta = \pi + \tan^{-1}(1) = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\]Thus, the polar form is:\[\frac{3\sqrt{2}}{2} \text{cis} \frac{5\pi}{4}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Form
Polar form is a way to express complex numbers, often making multiplicative operations more convenient. A complex number in polar form is represented as \( r(\cos \theta + i \sin \theta) \) or \( r \text{cis} \theta \), where:
- \( r \) is the modulus, or the absolute value, of the complex number. It represents the distance of the point from the origin in the complex plane.
- \( \theta \) is the argument of the complex number, giving the direction of the point in radians with respect to the positive real axis.
Complex Conjugate
The complex conjugate of a complex number is a key concept when simplifying expressions involving complex numbers.For any complex number \( a + bi \), its conjugate is \( a - bi \).
- It essentially reflects the complex number across the real axis.
- Using the conjugate helps in rationalizing the denominator, as seen in the given exercise.
- The product of a complex number and its conjugate is always a real number, which simplifies calculations and expressions.
Modulus of a Complex Number
The modulus of a complex number is its absolute value.For a given complex number \( z = a + bi \), the modulus is calculated as \( |z| = \sqrt{a^2 + b^2} \).
- The modulus represents the length of the vector from the origin to the point \((a, b)\) in the complex plane.
- It is always a non-negative real number.
Argument of a Complex Number
The argument of a complex number is the angle it forms with the positive direction of the real axis.For a complex number \( z = a + bi \), the argument \( \theta \) is found using the arctangent function \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
- This angle is usually given in radians.
- Modify \( \theta \) based on the quadrant in which \( z \) lies.
