Finding Modulus and Argument for a Complex Number

In summary, to find the modulus of a complex number, you square both the real and imaginary parts, add them, and then take the square root. For the argument of a complex number, you can use the tangent function to find the angle the complex number makes with the x-axis. The principal value is a convention for choosing which angle to use.
  • #1
ChrisBaker8
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Homework Statement



Find the modulus and the principal value of the argument for the complex number [tex]\sqrt{3}[/tex] - i

The Attempt at a Solution



I know the modulus is just 'square both, add, and square root of sum', so r = [tex]\sqrt{2}[/tex], but I don't know how to find the second part. I know vaguely that the argument = [tex]\theta[/tex], but I don't know where to go from here.

Do I need to convert the complex number into polar or euler form?
 
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  • #2
What two numbers did you 'square both, add, and square root of sum' to get sqrt(2)? The complex number a+bi can be drawn as the hypotenuse of a right triangle in the complex plane with a horizontal leg of length a and a vertical leg of length b. Haven't you seen this picture? The argument is the angle the hypotenuse makes with the x-axis. So you have tan(argument)=b/a. Remember trig? Principal value is a convention for choosing which of several possible angles might satisfy the tangent equation. Look it up, hopefully it will come with a nice picture.
 
  • #3
okay, I think I get the argument now

for the modulus, I added [tex]\sqrt{3}[/tex] [tex]^{2}[/tex] and i[tex]^{2}[/tex] to get 3 - 1, then square rooted to get [tex]\sqrt{2}[/tex]

is that wrong?
 
  • #4
ChrisBaker8 said:
okay, I think I get the argument now

for the modulus, I added [tex]\sqrt{3}[/tex] [tex]^{2}[/tex] and i[tex]^{2}[/tex] to get 3 - 1, then square rooted to get [tex]\sqrt{2}[/tex]

is that wrong?
Yes. You square the real part, sqrt(3), and the imaginary part, -1, add them, then take the square root. The imaginary part is the coefficient of i.
 
  • #5
okay, thanks
 

Related to Finding Modulus and Argument for a Complex Number

1. What are complex numbers and why are they important in arguments?

Complex numbers are numbers that consist of both a real part and an imaginary part. They are important in arguments because they allow for the representation of quantities that cannot be described by real numbers alone, such as the square root of a negative number.

2. How are complex numbers used in scientific arguments?

Complex numbers are used in scientific arguments to represent and manipulate quantities in a way that is not possible with real numbers. They are commonly used in fields such as physics, engineering, and mathematics to model and solve complex problems.

3. Can complex numbers be divided?

Yes, complex numbers can be divided, just like real numbers. To divide complex numbers, we use the process of rationalizing the denominator, similar to how we would with fractions.

4. What is the relationship between complex numbers and the Argand plane?

The Argand plane is a geometric representation of complex numbers, where the real part is plotted on the x-axis and the imaginary part is plotted on the y-axis. This allows for a visual representation of complex numbers and their operations.

5. Are there any real-life applications of complex numbers?

Yes, there are many real-life applications of complex numbers. They are commonly used in electrical engineering to represent and analyze alternating current circuits. They are also used in signal processing, quantum mechanics, and many other fields of science and engineering.

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