Problem 4 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Prove that \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}=2\left[\left|z_{1}\right|^{2}+\right.\) \(\left.\left|z_{2}\right|^{2}\right]\)

Step-by-Step Solution

Answer

The identity is true; algebraic expansion and cancelling terms verify the equality.

Step 1: Use the Definition of Magnitude

Recall that for a complex number \( z = x + yi \), the magnitude is defined as \( |z| = \sqrt{x^2 + y^2} \). For any complex numbers \( z_1 \) and \( z_2 \), the formula \( |a+b|^2 = |a|^2 + |b|^2 + 2\operatorname{Re}(a\bar{b}) \) can be helpful.

Step 2: Apply Magnitude Properties

For any complex numbers \( z_1 \) and \( z_2 \), expand \(|z_1+z_2|^2\) using the property. It becomes:\[|z_1 + z_2|^2 = (z_1 + z_2)(\overline{z_1 + z_2}) = z_1\overline{z_1} + z_2\overline{z_1} + z_1\overline{z_2} + z_2\overline{z_2}\]This simplifies to \( |z_1|^2 + 2\operatorname{Re}(z_1\overline{z_2}) + |z_2|^2 \).

Step 3: Expand \(|z_1 - z_2|^2\)

Using a similar process as in Step 2, expand \(|z_1 - z_2|^2\):\[|z_1 - z_2|^2 = (z_1 - z_2)(\overline{z_1 - z_2}) = z_1\overline{z_1} - z_2\overline{z_1} - z_1\overline{z_2} + z_2\overline{z_2}\]This simplifies to \( |z_1|^2 - 2\operatorname{Re}(z_1\overline{z_2}) + |z_2|^2 \).

Step 4: Combine Results

Now, add the expressions from Steps 2 and 3:\[|z_1 + z_2|^2 + |z_1 - z_2|^2 = (|z_1|^2 + 2\operatorname{Re}(z_1\overline{z_2}) + |z_2|^2) + (|z_1|^2 - 2\operatorname{Re}(z_1\overline{z_2}) + |z_2|^2)\]The \( 2\operatorname{Re}(z_1\overline{z_2}) \) terms cancel out, leaving:\[2|z_1|^2 + 2|z_2|^2\]

Step 5: Conclude the Proof

Therefore, we have shown that:\[|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2(|z_1|^2 + |z_2|^2)\]This proves the original statement is true.

Key Concepts

Complex Conjugate

Understanding the concept of a complex conjugate can simplify many calculations involving complex numbers. A complex number is often written in the form of \( z = a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The complex conjugate of \( z \), denoted as \( \overline{z} \), is \( a - bi \). Essentially, you flip the sign of the imaginary component, keeping the real part the same.

This transformation is useful because certain operations become easier. For instance, when multiplying a complex number by its conjugate, you are left with only real components since the imaginary parts cancel out:

• The result of multiplying \( z \) and \( \overline{z} \) is \( z \overline{z} = (a + bi)(a - bi) \). Expanding this gives \( a^2 + b^2 \), a real number.
• Utilizing complex conjugates simplifies finding the magnitude of complex numbers.

Using the complex conjugate is particularly useful in the context of this exercise where we are interested in the squared magnitudes of sums and differences of complex numbers.

Magnitude of a Complex Number

The magnitude of a complex number \( z = a + bi \) is a measure of its size or distance from the origin when plotted on the complex plane. It is computed as \( |z| = \sqrt{a^2 + b^2} \). Essentially, it's like using the Pythagorean theorem to find the hypotenuse of a right triangle formed by the real and imaginary parts.

For a problem involving sums and differences of complex numbers, understanding the magnitude helps us analyze the equation:

• Apply the identity: \( |a+b|^2 = |a|^2 + |b|^2 + 2\operatorname{Re}(a\overline{b}) \). This expands the squared magnitude by introducing the real part of the product \( a\overline{b} \).
• This becomes essential in troubleshooting when calculating the magnitude of \( z_1 + z_2 \) or \( z_1 - z_2 \).

In this particular exercise, separating the calculations for these magnitudes is crucial to see how the equation comes together.

Properties of Complex Numbers

Complex numbers obey a set of properties similar to real numbers, but with extra considerations for their imaginary parts. These properties allow for flexible computations that can simplify many operations.

When you work with additions and subtractions of complex numbers as in the exercise, certain key properties are always at play:

• Commutative Property of Addition: \( z_1 + z_2 = z_2 + z_1 \).
• Associative Property of Addition: \( (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \).
• Distributive Property: \( a(b + c) = ab + ac \), where \( a, b, c \) can be complex numbers.

On top of these, knowing how conjugates interact with these properties is crucial. For example, for any complex number \( z \), the equality \( z\overline{z} = |z|^2 \) holds. This property simplifies proving statements like the one given in the exercise, showing how mathematical relationships remain consistent even within the context of complex numbers.