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In this video, we’re going to learn how to extend operations on complex numbers into multiplication.
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We’ll begin by looking at how to perform multiplication of a complex number first by real numbers and then by another complex number.
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We’ll then extend this to include deriving a general rule for squaring complex numbers and consider how this might help us to raise a complex number to exponents higher than two.
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And finally, we’ll learn how to apply these processes to help us solve equations.
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If you’ve been studying complex numbers for a little while now, you might be aware that operations on complex numbers are very similar if not at times identical to operations on algebraic expressions.
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In fact, multiplying complex numbers is just like multiplying algebraic expressions except that we remember that the letter 𝑖 is not a variable.
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𝑖 is of course the solution to the equation 𝑥 squared equals negative one.
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This means that 𝑖 squared is equal to negative one and in fact, we often say that 𝑖 is equal to the square root of negative one.
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So let’s begin by considering how we might multiply complex number of the form 𝑧 equals 𝑎 plus 𝑏𝑖 by a constant.
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Let’s call our constant 𝑐, where 𝑐 is a real number.
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𝑐 multiplied by 𝑧 — written simply as 𝑐𝑧 — is exactly the same as 𝑐 multiplied by the whole complex number 𝑎 plus 𝑏𝑖.
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And we add parentheses to show that this is the case.
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We recall the distributive property which allows us to multiply each part of the complex number by the real number 𝑐.
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In doing so, we see that 𝑐𝑧 is equal to 𝑐𝑎 plus 𝑐𝑏𝑖.
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And for clarity, this might sometimes be written as 𝑎𝑐 plus 𝑏𝑐𝑖.
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And this probably comes of no surprise.
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This is exactly what we would expect if we were to multiply any two-term algebraic expression by a real constant.
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Let’s now have a look at an example of how this might work.
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If 𝑟 equals negative five plus two 𝑖 and 𝑠 equals negative eight minus two 𝑖, find two 𝑟 plus three 𝑠.
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Here, we’ve been given two complex numbers 𝑟 and 𝑠.
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We want to find the sum of two 𝑟 and three 𝑠.
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So we’ll split the problem up and begin by working out two 𝑟 and three 𝑠 separately.
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Two 𝑟 is two multiplied by the complex number negative five plus two 𝑖.
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We’ll distribute these brackets by multiplying each part of the complex number by our constant two.
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Two multiplied by negative five is negative 10 and two multiplied by two 𝑖 is four 𝑖.
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And we see that two 𝑟 is equal to negative 10 plus four 𝑖.
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We’ll now repeat this process for three 𝑠.
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This time, we multiply each part of the complex number 𝑠 by the constant three.
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Three multiplied by negative eight is negative 24 and three multiplied by negative two 𝑖 is negative six 𝑖.
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And now that we know the complex numbers two 𝑟 and three 𝑠, we need to find their sum.
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That’s negative 10 plus four 𝑖 plus negative 24 minus six 𝑖.
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And remember to add two complex numbers, we simply add their real parts and then individually add their imaginary parts.
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Negative 10 plus negative 24 is negative 34 and four 𝑖 plus negative six 𝑖 is negative two 𝑖.
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So for the complex numbers given, two 𝑟 plus three 𝑠 is equal to negative 34 minus three 𝑖.
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Now, this is all fine and well.
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But what it if we’d actually been multiplying our complex numbers by a purely imaginary number?
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We’ve already seen that the distributive property is really helpful in multiplying complex numbers by a real constant.
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And in fact, we can use this property to multiply a complex number by a purely imaginary number.
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That’s a number of the form 𝑐𝑖, where 𝑐 is a real number and 𝑖 is an imaginary number, the solution to the equation 𝑥 squared equals negative one.
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This time we’re going to multiply a complex number 𝑎 plus 𝑏𝑖 by 𝑐𝑖.
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When we do, we get 𝑐𝑖 multiplied by 𝑎 plus 𝑏𝑖.
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𝑐𝑖 multiplied by 𝑎 is 𝑐𝑎𝑖 and 𝑐𝑖 multiplied by 𝑏𝑖 is 𝑐𝑏𝑖 squared.
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But since 𝑖 squared equals is equal to negative one, we can write this as negative 𝑐𝑏.
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And if we instead write this in complex number form, we see that our complex number multiplied by a purely imaginary number is negative 𝑐𝑏 plus 𝑐𝑎𝑖.
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Now we have developed a formula for multiplying a complex number by a purely imaginary number.
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We should really focus on applying the processes each time rather than trying to learn these by heart.
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Here, we’re going to consider an example of where we can apply these processes to multiply a complex number by a purely imaginary number.
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What is negative seven 𝑖 multiplied by negative five plus five 𝑖?
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We have a complex number negative five plus five 𝑖 and we want to multiply it by a purely imaginary number negative seven 𝑖.
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And we know that multiplying complex numbers is just like multiplying algebraic expressions.
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Here, we can apply the distributive property for expanding brackets.
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We multiply each part inside the bracket by the number on the outside.
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That’s negative seven 𝑖 multiplied by negative five which is 35𝑖 and negative seven 𝑖 multiplied by five 𝑖 which is negative 35𝑖 squared.
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And here, we recall the fact that 𝑖 is the solution to the equation 𝑥 squared equals negative one such that 𝑖 squared must be equal to negative one.
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So negative 35𝑖 squared is the same as negative 35 multiplied by negative one which is simply 35.
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And since we now have a complex number which is of course a result of adding a real and a purely imaginary number, we write it as 35 plus 35𝑖.
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Now as we might expect, we can extend these ideas into multiplying two complex numbers.
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And we’ll begin by considering the general product of two complex numbers.
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Let’s say we have two complex numbers 𝑧 one and 𝑧 two such that 𝑧 one is equal to 𝑎 plus 𝑏𝑖 and 𝑧 two is equal to 𝑐 plus 𝑑𝑖.
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Their product 𝑧 one 𝑧 two is the product 𝑎 plus 𝑏𝑖 and 𝑐 plus 𝑑𝑖.
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And we’ve already seen that we can apply algebraic techniques to complex numbers.
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Here we can use any technique we like for multiplying two binomials.
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FOIL method and the grid method are two common methods.
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We’re going to look at the FOIL method.
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F stands for first.
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We multiply the first term in the first bracket by the first term in the second bracket.
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𝑎 multiplied by 𝑐 is simply 𝑎𝑐.
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O stands for outer.
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We multiply the outer terms and we get 𝑎𝑑𝑖.
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I stands for inner.
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We multiply the inner terms and we get 𝑏𝑐𝑖.
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And L stands for last.
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We multiply the last term in each bracket which is 𝑏𝑑𝑖 squared.
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And since 𝑖 squared is equal to negative one, we can write this last part as negative 𝑏𝑑.
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And we can rearrange this slightly and we see that the products of 𝑧 one and 𝑧 two is 𝑎𝑐 minus 𝑏𝑑 plus 𝑎𝑑 plus 𝑏𝑐𝑖.
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It’s a complex number with a real part 𝑎𝑐 minus 𝑏𝑑 and an imaginary part 𝑎𝑑 plus 𝑏𝑐.
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Now once again, we have developed a formula for multiplying a complex number by another complex number.
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But we should really focus on applying the processes each time.
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Multiply negative three plus 𝑖 by two plus five 𝑖.
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Multiplying two complex numbers is just like multiplying two binomials and we can use any technique we like.
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Let’s try the grid method.
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Two multiplied by negative three is negative six and two multiplied by 𝑖 is two 𝑖.
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Five 𝑖 multiplied by negative three is negative 15 𝑖 and five 𝑖 multiplied by 𝑖 is five 𝑖 squared.
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And of course, 𝑖 squared is equal to negative one.
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So five 𝑖 squared is five multiplied by negative one which is negative five.
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We’re going to simplify in a moment by collecting like terms.
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But currently, adding each part, we get negative six minus five plus two 𝑖 minus 15𝑖.
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Negative six minus five is negative 11 and two 𝑖 minus 15𝑖 is negative 13𝑖.
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So when we multiply negative three plus 𝑖 by two plus five 𝑖, we get negative 11 minus 13𝑖.
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In our next example, we’ll look at how we’ll extend these ideas into squaring complex numbers.
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If 𝑟 is equal to negative two plus four 𝑖 and 𝑠 is equal to eight minus 𝑖, find 𝑟 minus 𝑠 all squared.
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In this question, we’ve been given two complex numbers and we’re being asked to find the square of their difference, 𝑟 minus 𝑠.
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Now, we absolutely could write 𝑟 minus 𝑠 squared as 𝑟 minus 𝑠 multiplied by 𝑟 minus 𝑠 and expand these brackets as normal.
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When we do, we see that we have three unique parts: 𝑟 squared, 𝑠 squared, and negative two 𝑟𝑠.
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It’s rather a lot of work for us to evaluate each of these complex numbers.
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Instead, we’ll find the difference between the terms first and then we’ll square them.
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𝑟 minus 𝑠 is negative two plus four 𝑖 minus eight minus 𝑖.
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And to subtract complex numbers, we subtract their real parts and their imaginary parts.
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Alternatively, we can think of this and a little like collecting like terms.
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Before we do that though, let’s distribute the second lot of brackets by multiplying each part inside the bracket by negative one.
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That gives us negative eight plus 𝑖.
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Negative two minus eight is negative 10 and four 𝑖 plus is five 𝑖.
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So 𝑟 minus 𝑠 is negative 10 plus five 𝑖.
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This means that 𝑟 minus 𝑠 squared is negative 10 plus five 𝑖 squared.
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But since squaring a number is just the same as multiplying it by itself, we write this as negative 10 plus five 𝑖 multiplied by negative 10 plus five 𝑖.
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And multiplying two complex numbers is just like multiplying binomials.
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We can use any technique we like.
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Let’s look at the FOIL method.
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We’ll begin by multiplying the first term in the first bracket by the first term in the second bracket.
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Negative 10 multiplied by negative 10 is 100 we multiply the outer terms: negative 10 multiplied by five 𝑖 is negative 50𝑖.
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And we get the same if we multiply the inner two terms.
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Finally, we multiply the last term in each bracket.
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And we get 25𝑖 squared.
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But since 𝑖 squared is equal to negative one, we can write this as 25 multiplied by negative one which is negative 25.
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100 minus 25 is 75.
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And negative 50 minus 50 is negative 100.
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So we get negative 100𝑖.
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And we can see that 𝑟 minus 𝑠 all squared is 75 minus 100𝑖.
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Now that we’ve seen an example of how to square a complex number, let’s extend this and derive the general form.
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Let’s say we have a complex number 𝑧 in the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers.
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𝑧 squared is 𝑎 plus 𝑏𝑖 all squared.
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But we know that we can square a complex number by multiplying it by itself and applying the same techniques we use for expanding brackets.
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We multiply the first term in each bracket and we get 𝑎 squared.
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When we multiply the outer terms, we get 𝑎𝑏𝑖.
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And when we multiply the inner terms, we once again get 𝑎𝑏𝑖.
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And when we multiply the last terms, we get 𝑏 squared 𝑖 squared.
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But of course, 𝑖 squared is equal to negative one.
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So this last term is negative 𝑏 squared.
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And we see that 𝑧 squared is equal to 𝑎 squared minus 𝑏 squared plus two 𝑎𝑏𝑖.
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And we can say that in general when we square a complex number 𝑧 in the form 𝑎 plus 𝑏𝑖, the real part of 𝑧 squared is 𝑎 squared minus 𝑏 squared and the imaginary part is two 𝑎𝑏.
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We’ve seen throughout this video though that remembering the technique is more important than remembering the formula.
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In this case, learning the formula for a square of a complex number can actually be extremely useful.
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Let’s see an example of where it might help us to simplify a calculation.
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Find the real part of seven minus two 𝑖 all squared.
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Here, we’ve been given a complex number seven minus two 𝑖 for which we’re being asked to find the real part of its square.
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We could begin by writing seven minus two 𝑖 squared as seven minus two 𝑖 multiplied by seven minus two 𝑖 and then expanding the brackets fully.
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And we could use any technique for multiplying binomials.
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For example, we could use the FOIL method.
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We multiply the first term in the first bracket by the first term in the second bracket.
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That’s seven multiplied by seven which is 49.
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We multiply the outer terms.
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That’s seven multiplied by negative two 𝑖 which is negative 14𝑖.
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And we get the same number when we multiply the inner terms.
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We can multiply the last term in each bracket and we get positive four 𝑖 squared.
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But of course, 𝑖 squared is equal to negative one.
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So this simplifies somewhat to 49 minus 28𝑖 minus four which is 45 minus 28𝑖.
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Now for a complex number of the form 𝑎 plus 𝑏𝑖, its real part is 𝑎 and its imaginary part is 𝑏.
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And in this case, we can say that the real part of our complex number is 45.
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Now while this method is perfectly valid, the amount of work is a little unnecessary.
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Instead, we’re going to recall the general form for the square of a complex number 𝑎 plus 𝑏𝑖.
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It’s given by the formula 𝑧 squared equals 𝑎 squared minus 𝑏 squared plus two 𝑎𝑏𝑖.
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The real part is 𝑎 squared minus 𝑏 squared and the imaginary part is two 𝑎𝑏.
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Now in our complex number, the real part is seven; 𝑎 is equal to seven.
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And the imaginary part 𝑏 is equal to negative two.
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So we can say that the real part of 𝑧 squared is seven squared minus negative two squared or 49 minus four.
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49 minus four is equal to five.
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And that’s the same answer we worked out earlier.
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The rules we’ve learned for squaring complex numbers can actually also help us find higher powers of these numbers.
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If 𝑟 is equal to two plus 𝑖, express 𝑟 cubed in the form 𝑎 plus 𝑏𝑖.
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We’ll begin here by rewriting 𝑟 cubed slightly.
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𝑟 cubed is the same as 𝑟 times 𝑟 times 𝑟 which is the same as two plus 𝑖 times two plus 𝑖 times two plus 𝑖.
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We’ll begin by multiplying two plus 𝑖 by two plus 𝑖.
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And we could expand these two brackets just like expanding binomials.
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Alternatively, we recall that for complex number 𝑎 plus 𝑏𝑖, the square of that complex number is 𝑎 squared minus 𝑏 squared plus two 𝑎𝑏𝑖.
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The real part of our complex number 𝑎 is two.
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And the imaginary part is the coefficient of 𝑖; it’s one.
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We can square this number then by substituting 𝑎 is equal to two and B is equal to one into that formula.
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We get two squared minus one squared plus two times two times one 𝑖 and that’s all multiplied by two plus 𝑖.
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Two squared is four and one squared is one.
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So two squared minus one squared is three and two times two times one is four.
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So we have three plus four 𝑖 multiplied by two plus 𝑖.
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And we can multiply these brackets by using the FOIL method.
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We multiply the first term in each bracket.
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Three times two is six.
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We multiply the outer terms to get three 𝑖.
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And for the inner terms, it’s four 𝑖 multiplied by two which is eight 𝑖.
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We then multiply the last terms four 𝑖 multiplied by 𝑖 is four 𝑖 squared.
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And then since we know that 𝑖 squared is equal to negative one, four 𝑖 squared becomes negative four.
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And we also know that three 𝑖 plus eight 𝑖 is 11𝑖.
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So we see that our expression simplifies to two plus 11𝑖.
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So 𝑟 cubed in the form required is two plus 11𝑖.
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Now there are techniques that exist to simplify this process for higher powers of 𝑧, which we’ll learn about as we become more confident working with complex numbers.
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Our last example though will demonstrate how to solve an equation using complex numbers.
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Solve the equation 𝑖𝑧 is equal to negative four plus three 𝑖.
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Usually, we look to apply the rules for solving algebraic expressions.
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Here that will be dividing by 𝑖.
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But there is another technique we can use.
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We know that 𝑖 squared is negative one.
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So we’re going to multiply both sides of this equation by 𝑖.
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We then distribute the brackets by multiplying each bit inside the bracket by 𝑖.
00:17:03.920 --> 00:17:10.040
And since 𝑖 squared is negative one, we get negative 𝑧 equals negative four 𝑖 minus three 𝑖.
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We’ll multiply through by negative one and we see that 𝑧 is equal to three plus four 𝑖.
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And it’s always sensible to check our answers by substituting it back into the original equation: 𝑖 multiplied by three plus four 𝑖 is three 𝑖 plus four 𝑖 squared.
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And since four 𝑖 squared is four multiplied by negative one, we get negative four.
00:17:29.960 --> 00:17:33.160
And this is of course the same as negative four plus three 𝑖.
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In this video, we’ve learned that we can multiply two complex numbers using standard methods such as the grid method or FOIL method.
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We’ve seen that the square of a complex number 𝑎 plus 𝑏𝑖 is 𝑎 squared minus 𝑏 squared plus two 𝑎𝑏𝑖.
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And we also saw how we can use these techniques for higher powers of 𝑧 though this isn’t necessarily the most efficient method.