Video Transcript
In which quadrant of the argand plane does the complex number seven plus nine 𝑖 over three minus four 𝑖 lie?
The argand plane looks a lot like the Cartesian plane, but the horizontal and vertical axes have different meanings. We use the horizontal axis on an argand plane to represent the real part of a complex number. And we use the vertical axis to represent the imaginary part. Then, for example, the complex number 𝑧 equals three plus four 𝑖, which has a real part of three and an imaginary part of four, would be plotted at the coordinates three, four. That’s somewhere around here in the first quadrant of the argand diagram.
In order to determine which of the four quadrants this complex number seven plus nine 𝑖 over three minus four 𝑖 lies in, we need to consider the signs of its real and imaginary parts. Now, we can’t actually do this at the moment because our complex number has not been given in the general form 𝑧 equals 𝑎 plus 𝑏𝑖. So, we need to consider how we can manipulate this complex number first in order to bring it into the standard form.
There’s a trick that we can use to do this. And it’s a little like when we rationalize the denominator in a surd. We want to make the denominator of this quotient a real number. So, its imaginary part must be zero. And to do this, we multiply by the complex conjugate of the denominator. Recall that the complex conjugate of the general complex number 𝑎 plus 𝑏𝑖 is the complex number 𝑎 minus 𝑏𝑖. We simply changed the sign of the imaginary part, so the complex conjugate of three minus four 𝑖 is three plus four 𝑖.
However, we can’t just multiply the denominator of our quotient by this complex number because that would be changing the quotient’s value. We also need to multiply the numerator by the same quotient so that, overall, we’re multiplying by one and, therefore, finding an equivalent value. So, we now have the expression seven plus nine 𝑖 multiplied by three plus four 𝑖 over three minus four 𝑖 multiplied by three plus four 𝑖. We need to distribute each set of parentheses. And we’ll begin in the denominator so that we can remind ourselves why we multiply by the complex conjugate.
So, in the denominator then, multiplying the first terms together gives nine. Multiplying the outer terms together, three multiplied by four 𝑖 is 12𝑖. Multiplying the inner terms together gives negative 12𝑖. And then, multiplying the last terms together gives negative 16𝑖 squared. Now, we see that in the center of our expansion, we have plus 12𝑖 minus 12𝑖. And these two terms cancel each other out.
We also recall the key fact that 𝑖 squared is equal to negative one. So, we have nine minus 16 multiplied by negative one. That’s nine plus 16, which is 25. And the key point is that by multiplying three minus four 𝑖 by its complex conjugate three plus four 𝑖, we have got a real number. The imaginary part of the result is zero. This is an illustration of the general result that if we take a complex number 𝑎 plus 𝑏𝑖 and multiply it by its complex conjugate 𝑎 minus 𝑏𝑖, we will obtain a real number 𝑎 squared plus 𝑏 squared. We can see that, in our case, we obtain the number 25, which is equal to three squared plus four squared.
So, we now have a real number in the denominator of our quotient. We now need to distribute the parentheses in the numerator. Doing so gives 21 plus 28𝑖 plus 27𝑖 plus 36𝑖 squared. Recalling again that 𝑖 squared is equal to negative one, this simplifies to 21 plus 55𝑖 plus 36 multiplied by negative one. That’s 21 minus 36, which is negative 15, plus 55𝑖.
So, by multiplying both the numerator and denominator of this quotient by the complex conjugate of the denominator, we’ve found that this complex number is equivalent to the complex number negative 15 plus 55𝑖 over 25. Remember, the purpose of us doing this was so we could determine the signs of the real and imaginary parts of our complex number. So, by separating the real and imaginary parts up into to two separate fractions and then simplifying, we have that our complex number is equivalent to negative three-fifths plus eleven-fifths 𝑖.
So, we find that the real part of our complex numbers 𝑧 is negative three-fifths, which is negative. And the imaginary part of our complex number 𝑧 is eleven-fifths, which is positive. Our complex number will, therefore, be plotted on the argand plane using a negative value on the real axis and a positive value on the imaginary axis, which means it will be in the second quadrant.
So, by first finding an equivalent expression for our complex number by multiplying both the numerator and denominator by the complex conjugate of the denominator. And then, considering the signs of its real and imaginary parts, we’ve found that the complex number seven plus nine 𝑖 over three minus four 𝑖 lies in the second quadrant of the argand plane.
