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Find the complex conjugate of the complex number one plus π and the product of this number with its complex conjugate.
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So, thereβre two parts to this question.
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Firstly, weβre asked to find the complex conjugate of this complex number one plus π.
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Well, we can recall that the complex conjugate of a complex number is the complex number we get when we simply change the sign of its imaginary part.
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So, in general, the complex conjugate of the complex number π§ equals π plus ππ is the complex number π§ star, which is equal to π minus ππ.
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Weβve changed the sign of the complex part.
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Itβs no longer positive π.
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Itβs now negative π.
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So if we let π§ be our complex number, one plus π, then to find its complex conjugate π§ star, we simply change the sign of the imaginary part.
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So previously, we had plus π, which is plus one π.
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And we change it to negative π or negative one π.
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The complex conjugate of one plus π is therefore one minus π.
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The second part of this question asks us to find the product of this number.
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So thatβs our original complex number with its complex conjugate.
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So weβre looking for the product of π§ and π§ star.
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As weβve just found the complex conjugate to be one minus π, weβre therefore looking for the product of one plus π and one minus π.
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We can go ahead and distribute the parentheses.
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One multiplied by one gives one.
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And then, one multiplied by negative π gives negative π.
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π multiplied by one gives positive π.
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And then, π multiplied by negative π gives negative π squared.
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So we have one minus π plus π minus π squared.
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Now, of course, in the centre of our expression, negative π plus π simplifies to zero.
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So these two terms cancel out.
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And weβre left with one minus π squared.
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We need to recall here that π squared is equal to negative one.
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We therefore have one minus negative one or one plus one, which is equal to two.
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And so we found that the product of our complex number with its complex conjugate is two.
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In fact, there is actually a general result that we couldβve used here, which is that, for the complex number π§ equals π plus ππ, the product of π§ with its complex conjugate π minus ππ will always be equal to π squared plus π squared.
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We can see that this is certainly the case for our complex number one plus π.
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Both the real and imaginary parts are equal to one.
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And one squared plus one squared is equal to one plus one, which is equal to two.
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To see why this is the case, we just need to distribute the parentheses in the product π plus ππ multiplied by π minus ππ.
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And we see that, in the general case, just as it did in our specific example, the imaginary parts of this expansion cancel, leaving π squared minus π squared π squared.
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Thatβs π squared minus π squared multiplied by negative one, which is π squared plus π squared.
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So weβve completed the problem.
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The complex conjugate of one plus π is one minus π.
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And the product of one plus π with its complex conjugate is two.