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Chapter 6: Problem 45

Show that addition of complex numbers is associative, meaning that $$ u+(w+z)=(u+w)+z $$ for all complex numbers \(u, w,\) and \(z\).

Short Answer

Expert verified
To show that addition of complex numbers is associative, we start by representing three complex numbers u, w, and z as \(u = a + bi\), \(w = c + di\), and \(z = e + fi\), where a, b, c, d, e, and f are real numbers. Adding w and z gives us \((w+z) = (c + e) + (d + f)i\). Adding u to the result, we get \(u+(w+z) = (a + (c + e)) + (b + (d + f))i\). Now, adding u and w gives us \((u+w) = (a + c) + (b + d)i\), and adding z to the result provides \((u+w)+z = ((a + c) + e) + ((b + d) + f)i\). Comparing the two expressions, we can see that the real and imaginary parts are equal, and therefore, the addition of complex numbers is associative: \(u+(w+z) = (u+w)+z\).

Step by step solution

01

Define complex numbers u, w, and z

Let u, w, and z be any three complex numbers. They can be represented in the form: \(u = a + bi\), \(w = c + di\), \(z = e + fi\), where a, b, c, d, e, and f are real numbers, and i is the imaginary unit with the property \(i^2 = -1\).
02

Add w and z

We first add the complex numbers w and z using the definition of complex number addition. This involves adding the real parts together and adding the imaginary parts together separately: \((w+z) = (c + di) + (e + fi) = (c + e) + (d + f)i\).
03

Add u to (w+z)

Now, we add u to the sum of w and z, computed in the previous step: \(u+(w+z) = (a + bi) + ((c + e) + (d + f)i) = (a + (c + e)) + (b + (d + f))i\).
04

Add u and w

We'll now add u and w: \((u+w) = (a + bi) + (c + di) = (a + c) + (b + d)i\).
05

Add (u+w) to z

Next, we'll add the sum of u and w to z: \((u+w)+z = ((a + c) + (b + d)i) + (e + fi) = ((a + c) + e) + ((b + d) + f)i\).
06

Compare u+(w+z) and (u+w)+z

Now, we'll compare the expressions we found for u+(w+z) and (u+w)+z: \(u+(w+z) = (a + (c + e)) + (b + (d + f))i\), \((u+w)+z = ((a + c) + e) + ((b + d) + f)i\). It's clear that the real parts of both expressions are equal: \((a + (c + e)) = ((a + c) + e)\) and the imaginary parts are also equal: \((b + (d + f)) = ((b + d) + f)\). Since each part of the complex numbers is equal, we can conclude that the addition of complex numbers is associative: \(u+(w+z) = (u+w)+z\).

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