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Modulus of a Complex Number

Q: How can any complex number ( z = a + bi ) be represented in the Argand Diagram to find its distance from the origin?

- Let point P(a, b) represent ( a + bi ) in the plane - Draw perpendiculars PM and PL on Re-axis and Im-axis - Let OM = a, MP = b - Find the distance of P from the origin as OP

Q: What is the notation used to denote the modulus of any complex number z such that ( z = a + bi )?

- The modulus of z such that ( z = a + bi ) is denoted by |z| - Express |z| as the square root of the sum of squares of a and b

9 important questions on Modulus of a Complex Number

How can any complex number \( z = a + bi \) be represented in the Argand Diagram to find its distance from the origin?

- Let point P(a, b) represent \( a + bi \) in the plane
- Draw perpendiculars PM and PL on Re-axis and Im-axis
- Let OM = a, MP = b
- Find the distance of P from the origin as OP

What is OP called in this context, and how is it calculated?

- OP is called the modulus or absolute value of the complex number \( a + bi \)
- Calculate OP as the square root of the sum of squares of a and b

What is the notation used to denote the modulus of any complex number z such that \( z = a + bi \)?

- The modulus of z such that \( z = a + bi \) is denoted by |z|
- Express |z| as the square root of the sum of squares of a and b

How is the expression for the modulus of a complex number z, in terms of a and b, formally represented?

- The modulus of z, expressed as |z|, is the square root of the sum of squares of a and b
- It is formally represented as the square root of \(a^2 + b^2\)

Can the modulus of any complex number z be calculated if the real and imaginary parts a and b are known?

- Yes, the modulus of z can be determined using the formula |z| = \(\sqrt{a^2 + b^2}\)
- Knowing the values of a and b allows for the calculation of |z| easily

Find the modulus of the following complex numbers:

- 5 + 2i
- 4 - 3i
- 1 + 2i

For the complex numbers z1 and z2, show that:

i. z1z2 = z2z1
ii. z1/z2 = z1/z2
iii. |z1+z2| ≤ |z1| + |z2|
iv. |z1-z2| ≥ ||z1| - |z2||

Express the complex number (3 + 4i) / (4 - 3i) in the form a + bi.

-3/7 + 25/7i

If (a+bi)(x+yi) = (a-bi) / (x+yi) and (1+i)(x-yi) = (x+yi), show that:

- a = x
- b = y

The question on the page originate from the summary of the following study material:

LINEAR ALGEBRA FOE

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Modulus of a Complex Number

9 important questions on Modulus of a Complex Number

How can any complex number \( z = a + bi \) be represented in the Argand Diagram to find its distance from the origin?

What is OP called in this context, and how is it calculated?

What is the notation used to denote the modulus of any complex number z such that \( z = a + bi \)?

How is the expression for the modulus of a complex number z, in terms of a and b, formally represented?

Can the modulus of any complex number z be calculated if the real and imaginary parts a and b are known?

Find the modulus of the following complex numbers:

For the complex numbers z1 and z2, show that:

Express the complex number (3 + 4i) / (4 - 3i) in the form a + bi.

If (a+bi)(x+yi) = (a-bi) / (x+yi) and (1+i)(x-yi) = (x+yi), show that:

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