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In the complex plane, the horizontal axis is called the real axis and the vertical axis is called the imaginary axis. The complex number a + bi graphed in the complex plane is comparable to the point (a, b) graphed in the standard (x, y) coordinate plane. The modulus of the complex number a + bi is given by:
\[\sqrt{a^2 + b^2}\]
Question:
Which of the complex numbers z1, z2, z3, z4 and z5 below has the greatest modulus?
Answer:
Okay, so what is this question really asking?
All it is asking is for you to find the largest value of the square root of the sum of the squares of our coordinate points, or find:
\[\sqrt{x^2 + y^2}\]
The challenge is that we are not given the real x and y values of the coordinate points. So we have to estimate the coordinate points of our z points.
Because we are working with squares, negatives are not a factor, so we can eliminate the smaller numbers as we are just taking a number, positive or negative and taking the square of it. Let us estimate our coordinates: z1 = (4, 5) z2 = (2, 1) z3 = (2, 3) z4 = (2, 2) z5 = (4, 2) We are looking for whichever point has the largest combination of its coordinate points. At a glance, the two points with the largest coordinates are z1 and z5. Let's find the modulus of z5, and smaller of the two.
\[\sqrt{4^2 + (2)^2}\]
\[\sqrt{16 + 4}\]
\[\sqrt{20}\]
\[4.5\]
And the modulus of z1:
\[\sqrt{(4)^2 + 5^2}\]
\[\sqrt{16 + 25}\]
\[\sqrt{41}\]
\[6.4\]
We can see that the modulus of z1, 6.4, is higher than that of z5.
Final answer: F, z1.
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