WEBVTT
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Given that π§ equals eight plus four π, find the modulus of π§.
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Well, to enable us to find the modulus of our complex number, what we need to do is actually consider a rule.
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And the rule is that for a complex number in the form π§ equals π plus ππ, its modulus is found by the equation: the modulus of the complex number equals the square root of π squared plus π squared.
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And just to help us remember what the form π§ equals π plus ππ actually means, well, it actually means where we have the complex number split into different parts.
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So we have the Re π§ is π and thatβs our real part.
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And then, we have π and π is Im π§, which is the imaginary part of our complex number.
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Okay, great, so now we have this formula for finding the modulus.
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We can do that and solve the problem.
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Okay, given that our complex number is equal to eight plus four π, then eight is actually going to be our π.
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And our positive four β thatβs because weβve got a plus sign in front of our four β is going to be our π.
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Okay, so we can now substitute these into our equation for the modulus.
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So weβre gonna get the modulus of π§ is equal to the square root of and then eight squared because thatβs our π and then plus four squared because four is our π.
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Okay, great, so now letβs calculate this.
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And this gives us that the modulus of π§ is equal to the square root of 64 plus 16.
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So therefore, the modulus of π§ is equal to root 80.
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Okay, so is this the final answer?
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Well, actually, if we get to this point and weβve actually got a surd, what we always say is trying to simplify the surd where we can.
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And weβre actually gonna use this surd rule to actually help us simplify our surd root 80 because we know that root ππ is equal to root π multiplied by root π.
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And what we want to do is actually find the highest square number.
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Thatβs one of the factors of 80.
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So therefore, weβre going to get the modulus of π§ is equal to root 16 root five.
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Thatβs because 16 multiplied by five is 80 sounds like our ππ.
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But also 16 is actually the highest square number that goes into 80.
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So therefore, we can say that the modulus of π§ is going to be equal to four root five.
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And thatβs because the square root of 16 is four.
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So it gives us four root five.
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And we actually found that by using the formula that the modulus of a complex number is equal to square root of π squared plus π squared, where the complex number is in the form π§ is equal to π plus ππ.