(a)

Draw the point (7, 4) on the Argand diagram

(b)

Modulus of Z
= sqrt (7² + 4²)
= sqrt 65

Argument of Z
= tan¯¹ (4/7)
= 0.5191461142 rad
~ 0.519 rad

(Does this appear similar to the ideas you used in R-formula?)

(c)

B = X/Z²

I use |A| for modulus of A.

|B|
= | X/Z² |
= |X| / |Z²|
= |X| / |Z|²
= 4 / |sqrt 65|²
= 4/65

I use arg (A) for argument of A.

arg B
= arg (X/Z²)
= arg X - arg Z² (sounds like logarithm a little?)
= arg X - 2 arg Z
= π/2 - 2 (0.5191461142)
= 0.5325040983
~ 0.533 rad

[Note that here X = 4 on the imaginary side, so this is basically on the positive side of the vertical axis; think of Sec 3 A Maths Trigo, if the "east" is 0 degrees, then the "north" is 90 degrees]

In exponential form,

B = (4/65) e^(0.533j)

i dont understand what is going on for the whole of part c tho, what i did was B=(4/(7+4j)^2) j=(4/33+56j)j .then i couldnt continue as there is no j

I did look into that approach at first, but I disregarded it later. The approach still continues to work.

The “j” outside can be brought into the numerator. Then, we “rationalise” the denominator (after all, j is the “square root” of a number, just like surds for real numbers).

…
= (4j) / (33 + 56j)
= (4j) (33 - 56j) / [(33 + 56j) (33 - 56j)]
= …

But doing what I did (using a set of modulus and argument rules for complex numbers) would be more straightforward in this context.

i dont understand what is going on for the modulus and argument part tho, why is modulus needed when they say find the susceptance B (instead of |B|) and why is argument used? sorry complex number is really new to me so i don't really know what is going on

also, sorry for the late reply, didnt receive any notification from the app (only saw it when i open the app)

Oh, forgot to respond last week.

I suggest you go through the JC level notes on H2 Math's Complex Numbers. You can find them from some online resources. Though I'm certain poly courses use "j" instead of "i".

Generally in solving complex numbers, there are three formats in expressing our complex number.

First format is the standard format x + jy where j is your imaginary number sqrt -1.

Second format is the polar form, also known as the modulus-argument form. We write this as R cos θ + R (j sin θ) or simply R (cos θ + j sin θ).

[Try to recall the derivation of the R-formula; do the expressions R cos θ and R sin θ ring a bell? Where did they appear from?]

Third format is the exponential form Re^(jθ) which is pretty much equivalent to the polar form.

Suppose you have a complex number Z = (1 + j) / (1 - 2j). How would you express the complex number?

Here, there are several ways to proceed.

One, we rationalise the expression before finding the modulus and the argument.

Two, we find the modulus and argument of 1 + j first, followed by the modulus and argument of 1 - 2j, before we combine the two modulus together and the two arguments together using rules which appear similar to logarithms.

The working which I have done earlier for your question involves this second approach.

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"i dont understand what is going on for the modulus and argument part tho, why is modulus needed when they say find the susceptance B (instead of |B|) and why is argument used? sorry complex number is really new to me so i don't really know what is going on"

So in this case, we wish to find the susceptance B in terms of complex numbers, but we would need to extract them from expressions. We have no choice but to define B using its modulus and its argument, both of which we would have to obtain beforehand.

thanks! by the way for complex numbers polar and exponential form, do we take negative when finding the angle? since for R formula we do not take negative (ignore the negative sign in numbers) do we apply it in complex numbers as well?

"thanks! by the way for complex numbers polar and exponential form, do we take negative when finding the angle? since for R formula we do not take negative (ignore the negative sign in numbers) do we apply it in complex numbers as well?"

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The way we call angles in polar and exponential form is largely similar to the rules of trigonometric solving which you have learned. Basic angles etc. The same rules apply as we find arguments/angles in complex numbers.

The R-formula was designed to provide several forms so as to cover all the possibilities (hence we use positive values for a and b), but in general we only call values positive when the different formats have been clearly outlined (like the R-formula).

ohh but its confusing though, as for polar and exponential form, when i take positive sometimes my answer is correct, sometimes its wrong. then when my teacher goes through it, i saw him putting in the negative sign too, but sometimes he dont. so i dont know which is correct anymore...

by the way to use the de moivre's theorem, must the angle in polar form be in radian or can it be in degree too?

It depends on the quadrants.

For example, (1, -2) is a point in the fourth quadrant. Then, your angle will be that of a fourth-quadrant version.

For complex numbers, always put your answers in radian form. This is because radian measure for angles is kind of a “length equivalent” for degree angles. We can add a length to another length but not a length to an angle, which is why we use radians.

ohhh the angle will always be relative to the the positive x-axis right and between 0 to 180/-180degrees? eg fourth quad will always be a negative angle, third quadrant will be between - 90 to -180degrees?

to find 2nd and 3rd quadrant angle do we always take +ve to find basic angle then use 180-basic angle?

O Level does it as 0 to 2pi (0 to 360).

Complex number does it as -pi to pi (-180 to 180).

Apart from this, the operations are pretty much the same.

Recall from O Level how each “main” angle theta is related to its basic/reference acute angle alpha.

For angles within 0 to 2pi,
1st: alpha
2nd: pi - alpha
3rd: pi + alpha
4th: 2pi - alpha

Complex number has its limits within -pi and pi instead, so the third quadrant and fourth quadrant values earlier would have to be subtracted by 2pi (angle goes back one cycle) to meet this new set of limits.

1st: alpha (no change)
2nd: pi - alpha (no change)
3rd: (pi + alpha) - 2pi = -pi + alpha
4th: (2pi - alpha) - 2pi = -alpha