Problem 1 - Complex numbers from A to...Z

Question

Let \(z_{1}, z_{2}, z_{3}\) be complex numbers such that $$ \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=r>0 $$ and \(z_{1}+z_{2}+z_{3} \neq 0\). Prove that $$ \left|\frac{z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}}{z_{1}+z_{2}+z_{3}}\right|=r $$

Step-by-Step Solution

Answer

Question: Prove that if $|z_1| = |z_2| = |z_3| = r > 0$ and $z_1 + z_2 + z_3 \neq 0$, then $|\frac{z_1z_2 + z_2z_3 + z_3z_1}{z_1 + z_2 + z_3}| = r$. Answer: To prove this, we followed these steps: 1. Declared the relationship between magnitudes and complex numbers as $|z_1| = |z_2| = |z_3| = r > 0$. 2. Used the properties of magnitudes to rewrite the equation as $\frac{|z_1z_2| + |z_2z_3| + |z_3z_1|}{|z_1 + z_2 + z_3|}$. 3. Applied the given relationship between magnitudes to get $\frac{3r^2}{|z_1 + z_2 + z_3|}$. 4. Proved that the magnitude of the complex fraction is equal to $r$ by showing that $r = |z_1 + z_2 + z_3|$.

Step 1: Declare the relationship between magnitudes and complex numbers

Given that the magnitude of each complex number \(z_1, z_2, z_3\) is equal to a positive real number \(r\), we can write the relationship as: $$ \left|z_1\right|=\left|z_2\right|=\left|z_3\right|=r>0 $$

Step 2: Use the properties of magnitudes to manipulate the equation

We know that for complex numbers, the magnitude of a product is equal to the product of the magnitudes. So, we can rewrite the magnitude in the equation as a product: $$ \left|\frac{z_1 z_2+ z_2 z_3+ z_3 z_1}{z_1+ z_2+ z_3}\right|= \frac{\left|z_1 z_2\right|+ \left|z_2 z_3\right|+ \left|z_3 z_1\right|}{\left|z_1+ z_2+ z_3\right|} $$

Step 3: Apply the given relationship between magnitudes

Since we know that \(\left|z_1\right| = \left|z_2\right| = \left|z_3\right| = r\), we can now replace these values in our equation: $$ \frac{r^2 + r^2 + r^2}{\left|z_1+ z_2+ z_3\right|} = \frac{3r^2}{\left|z_1+ z_2+ z_3\right|} $$

Step 4: Prove that the magnitude of the complex fraction is equal to r

To prove that the magnitude of this complex fraction is equal to r, we have to show that $$ \frac{3r^2}{\left|z_1+ z_2+ z_3\right|} = r $$ Since \(3r^2 \neq 0\), we can divide both sides by \(3r\): $$ \frac{r}{\left|z_1+ z_2+ z_3\right|} = 1 $$ Multiplying both sides by \(\left|z_1+ z_2+ z_3\right|\), we get: $$ r = \left|z_1+ z_2+ z_3\right| $$ This proves that the magnitude of the complex fraction is equal to \(r\) as required.

Key Concepts

Magnitude of Complex Numbers

Understanding the magnitude of complex numbers is crucial in complex number algebra. A complex number, written in the form of a + bi, where a and b are real numbers and i is the imaginary unit, has a magnitude (or modulus) that reflects its distance from the origin in the complex plane.

Mathematically, the magnitude of a complex number z = a + bi is denoted by |z| and can be calculated using the formula:
\[ |z| = \sqrt{a^2 + b^2} \]
In our exercise, the magnitude of each complex number |z_1|, |z_2|, and |z_3| is given as r. This implies that each complex number is r units away from the origin, although their actual position on the complex plane may vary.

Properties of Magnitudes

The properties of magnitudes play an integral part in manipulating complex number expressions. One such property states that the magnitude of a product of complex numbers equals the product of their magnitudes. This can be expressed as:
\[ |z_1 z_2| = |z_1| \cdot |z_2| \]
Another important property is that the magnitude of a sum is less than or equal to the sum of magnitudes, known as the Triangle Inequality:
\[ |z_1 + z_2| \leq |z_1| + |z_2| \]
In our original exercise, by applying these properties correctly, we could rewrite the expression involving magnitudes and reach the conclusion that the magnitude of the given complex fraction equals r, demonstrating a practical application of these properties in solving complex number problems.

Complex Number Algebra

Complex number algebra involves the operations of addition, subtraction, multiplication, and division of complex numbers. The key to mastering complex number algebra is understanding how to combine these operations with the properties of complex numbers.

For example, when adding complex numbers, you combine the real parts and the imaginary parts separately:
\[ (a_1 + b_1i) + (a_2 + b_2i) = (a_1 + a_2) + (b_1 + b_2)i \]
The solution to our problem also relied on an understanding of how magnitudes of products and sums work. In complex algebra, it's essential to remember that the operations must respect the rules of both algebra and the intrinsic properties of complex numbers. This approach was used in the exercise to simplify the expressions and logically prove the required equality.