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Chapter 1: Problem 33

Find \(z \bar{z}\) and \(|z|\) when \(z=3+4 i\)

Short Answer

Expert verified
The short answer is: \(z\bar{z}= 25\) and \(|z|= 5\).

Step by step solution

01

Find the conjugate of the complex number

To find the conjugate of a complex number in the form \(z=a+bi\), we only change the sign of the imaginary part. So, given \(z=3+4i\), the conjugate is \(\bar{z} = 3-4i\)
02

Calculate \(z\bar{z}\)

Now that we have the conjugate of the complex number, we can multiply \(z\) and \(\bar{z}\). \[ z\bar{z}=(3+4i)(3-4i) \] Use the distributive property (FOIL method) to perform the multiplication: \[ (3\times 3) + (3\times -4i) + (4i\times3) + (4i\times -4i) \] Now, calculate the product of each term: \[ 9 - 12i + 12i - 16i^2 \] Since \(i^2=-1\): \[ 9 - 16(-1) = 9 + 16 = 25 \] So, \( z\bar{z} = 25 \)
03

Calculate the modulus \(|z|\)

The modulus of a complex number \(z=a+bi\) is given by the formula: \[ |z| = \sqrt{a^2 + b^2} \] For our given complex number \(z=3+4i\), plug in the values of \(a\) and \(b\) into the formula: \[ |z| = \sqrt{3^2 + 4^2} \] Calculate the square of each term: \[ |z| = \sqrt{9 + 16} \] Now, add the terms under the square root: \[ |z|= \sqrt{25} \] Finally, take the square root of 25, which is 5. So, \(|z|= 5\). Results: - \(z\bar{z}= 25\) - \(|z|= 5\)

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Most popular questions from this chapter

Let $\mathrm{V}=\left\\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in F\right\\}\(, where \)F\( is a field. Define addition of elements of \)\mathrm{V}$ coordinatewise, and for \(c \in F\) and $\left(a_{1}, a_{2}\right) \in \mathrm{V}$, define $$ c\left(a_{1}, a_{2}\right)=\left(a_{1}, 0\right) . $$ Is \(\mathrm{V}\) a vector space over \(F\) with these operations? Justify your answer.

Is the set of all differentiable real-valued functions defined on \(R\) a subspace of \(C(R)\) ? Justify your answer.

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Find the vector \(v\) identified with the directed line segment \(P Q\) for the points: (a) \(P(2,3,-7)\) and \(Q(1,-6,-5)\) in \(\mathbf{R}^{3}\) (b) \(P(1,-8,-4,6)\) and \(Q(3,-5,2,-4)\) in \(\mathbf{R}^{4}\)

Let \(W\) be a subspace of a (not necessarily finite-dimensional) vector space \(V\). Prove that any basis for \(W\) is a subset of a basis for \(V\).
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