Chapter 2: Problem 48
Prove that \(|z|=\sqrt{z^{*} z}\) for any complex number \(z\).
Short Answer
Expert verified
\(|z|=\sqrt{z^{*} z}\) since \(z^{*}z=a^2+b^2\) and \(|z|=\sqrt{a^2+b^2}\).
Step by step solution
01
Understanding Modulus of a Complex Number
Given a complex number \(z = a + bi\), its modulus \(|z|\) is defined as \(|z| = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
02
Conjugate of a Complex Number
The conjugate of \(z = a + bi\) is \(z^{*} = a - bi\). This flips the sign of the imaginary part of the complex number while keeping the real part the same.
03
Multiplication of z and its Conjugate
To find \(z^{*}z\), multiply \(z\) by its conjugate: \((a - bi)(a + bi) = a^2 + abi - abi - (bi)^2\). The middle terms cancel out, and since \(i^2 = -1\), you get \(a^2 + b^2\).
04
Relating Modulus to the Expression
The expression \(z^{*}z = a^2 + b^2\) matches the inside of the modulus definition \(|z| = \sqrt{a^2 + b^2}\). Therefore, taking the square root of both sides of \(z^{*}z\), we show that \(|z| = \sqrt{z^{*}z}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate of a Complex Number
Complex numbers are numbers that have both a real part and an imaginary part. A complex number is typically written as \(z = a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit, which satisfies \(i^2 = -1\). The conjugate of a complex number, noted as \(z^{*}\), is formed by changing the sign of the imaginary component. For the complex number \(z = a + bi\), its conjugate \(z^{*}\) would be \(a - bi\). This simply involves negating the imaginary part, leaving the real part unchanged.
Understanding conjugates is crucial in complex number arithmetic, especially when simplifying expressions or solving equations. Conjugates have a special property: when a complex number is multiplied by its conjugate, it results in a real number. This is because the imaginary parts cancel each other out, showing their importance in finding the modulus and in simplifying expressions.
Understanding conjugates is crucial in complex number arithmetic, especially when simplifying expressions or solving equations. Conjugates have a special property: when a complex number is multiplied by its conjugate, it results in a real number. This is because the imaginary parts cancel each other out, showing their importance in finding the modulus and in simplifying expressions.
Multiplication of Complex Numbers
Multiplying complex numbers takes a bit more work than multiplying real numbers. If you have two complex numbers, \(z_1 = a + bi\) and \(z_2 = c + di\), you multiply them by distributing each term just like polynomials:
When multiplying a complex number by its conjugate, using this method effectively reveals why it results in a real number, as in \((a + bi)(a - bi) = a^2 + b^2\), which is crucial for understanding the connection to modulus.
- First, multiply the real parts: \(ac\).
- Next, multiply the outer parts: \(adi\).
- Then, multiply the inner parts: \(bci\).
- Finally, multiply the imaginary parts: \(bdi^2\).
When multiplying a complex number by its conjugate, using this method effectively reveals why it results in a real number, as in \((a + bi)(a - bi) = a^2 + b^2\), which is crucial for understanding the connection to modulus.
Square Root and Modulus
The modulus of a complex number is a measure of its "size" or distance from the origin in the complex plane. For any complex number \(z = a + bi\), the modulus is computed as \(|z| = \sqrt{a^2 + b^2}\). This definition of modulus makes use of the Pythagorean Theorem, reflecting the distance from the point \((a, b)\) to the origin \((0, 0)\) in a Cartesian plane.
When you multiply the complex number by its conjugate as shown earlier, \(z^{*}z = a^2 + b^2\), you arrive at a real number which is precisely the square of the modulus. Therefore, \(|z| = \sqrt{z^{*}z}\). This relationship allows you to connect the algebraic and geometric interpretations of complex numbers, thereby proving the initial proposition. Hence, understanding modulus not only gives a number but also an insight into the geometric nature of complex numbers by associating them to magnitudes.
When you multiply the complex number by its conjugate as shown earlier, \(z^{*}z = a^2 + b^2\), you arrive at a real number which is precisely the square of the modulus. Therefore, \(|z| = \sqrt{z^{*}z}\). This relationship allows you to connect the algebraic and geometric interpretations of complex numbers, thereby proving the initial proposition. Hence, understanding modulus not only gives a number but also an insight into the geometric nature of complex numbers by associating them to magnitudes.