Let's find the modulus of the complex expression \( \rm z+\frac{2}{z} \) when \( z = 1 + i \).

First, we need to calculate the value of the expression \( \rm z+\frac{2}{z} \).

Calculating the Complex Expression z + 2/z

We are given \( z = 1 + i \).

Now, let's find \( \frac{2}{z} \):

\[ \frac{2}{z} = \frac{2}{1 + i} \]

To simplify this fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1 + i \) is \( 1 - i \).

\[ \frac{2}{1 + i} = \frac{2}{1 + i} \times \frac{1 - i}{1 - i} \]

Using the difference of squares formula \( (a+b)(a-b) = a^2 - b^2 \) in the denominator:

\[ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

So, the expression becomes:

\[ \frac{2(1 - i)}{2} = 1 - i \]

Now, we add \( z \) and \( \frac{2}{z} \):

\[ z + \frac{2}{z} = (1 + i) + (1 - i) \]

Combine the real parts and the imaginary parts:

\[ z + \frac{2}{z} = (1 + 1) + (i - i) = 2 + 0i = 2 \]

The complex expression \( \rm z+\frac{2}{z} \) simplifies to the real number 2.

Finding the Modulus of z + 2/z

We found that \( z + \frac{2}{z} = 2 \). Now we need to find the modulus of this result.

For a complex number \( a + bi \), its modulus is given by \( |a + bi| = \sqrt{a^2 + b^2} \).

In our case, the complex number is 2, which can be written as \( 2 + 0i \). Here, \( a = 2 \) and \( b = 0 \).

The modulus is:

\[ \left| z + \frac{2}{z} \right| = |2 + 0i| = \sqrt{2^2 + 0^2} \]

\[ = \sqrt{4 + 0} = \sqrt{4} = 2 \]

Therefore, the modulus of \( \rm z+\frac{2}{z} \) is 2.

Understanding Modulus of a Complex Number

The modulus of a complex number \( z = a + bi \) represents its distance from the origin \((0, 0)\) in the complex plane. It is a non-negative real number.

• If \( z = a + bi \), then \( |z| = \sqrt{a^2 + b^2} \).
• If the complex number is purely real, like \( z = a \), then \( |z| = |a| \).
• If the complex number is purely imaginary, like \( z = bi \), then \( |z| = |b| \).

In this problem, \( \rm z+\frac{2}{z} \) resulted in a purely real number, 2. The modulus of 2 is simply \( |2| \), which is 2.

Revision Table: Complex Number Basics

Additional Information on Complex Number Operations

Operating with complex numbers involves treating them somewhat like binomials, with the special rule that \(i^2 = -1\).

• Addition/Subtraction: Add/subtract the real parts and the imaginary parts separately: \( (a+bi) \pm (c+di) = (a \pm c) + (b \pm d)i \).
• Multiplication: Use the distributive property (FOIL) and substitute \(i^2 = -1\): \( (a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i \).
• Division: Multiply the numerator and denominator by the conjugate of the denominator: \( \frac{a+bi}{c+di} = \frac{a+bi}{c+di} \times \frac{c-di}{c-di} = \frac{(ac+bd) + (bc-ad)i}{c^2+d^2} \).

These operations are fundamental when working with complex numbers and their properties like modulus and argument.