If Z = 1 + i, where i = √-1, then what is the modulus of \(\rm z+\frac{2}{z}?\)
2
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.
| Step | Calculation | Result |
|---|---|---|
| Given z | \(z = 1+i\) | |
| Calculate 2/z | \( \frac{2}{1+i} = \frac{2(1-i)}{(1+i)(1-i)} = \frac{2(1-i)}{2} \) | \( \frac{2}{z} = 1-i \) |
| Calculate z + 2/z | \( (1+i) + (1-i) = (1+1) + (i-i) \) | \( z + \frac{2}{z} = 2 \) |
| Calculate Modulus | \( |2| = \sqrt{2^2 + 0^2} = \sqrt{4} \) | \( \left| z + \frac{2}{z} \right| = 2 \) |
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
| Concept | Description | Formula/Example |
|---|---|---|
| Complex Number | A number of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i = \sqrt{-1}\). | \(z = 3 + 4i\) |
| Real Part | The number \(a\) in \(a + bi\). | For \(3 + 4i\), real part is 3. |
| Imaginary Part | The number \(b\) in \(a + bi\) (not including \(i\)). | For \(3 + 4i\), imaginary part is 4. |
| Conjugate | For \(a + bi\), the conjugate is \(a - bi\). | Conjugate of \(1+i\) is \(1-i\). |
| Modulus | The magnitude or length of the complex number from the origin in the complex plane. | \(|a+bi| = \sqrt{a^2 + b^2}\) |
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.
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