# Homework Help: Square root of complex number on a calculator

1. Jun 6, 2015

### Gremlin

I'm working through some examples in a text book but i am unable to get the desired answer on my calculator, i keep getting math error and various other results which are not the answer i'm looking for.

What i have is:

√ 62.9∠88.2 / 0.00165∠72.3

Please could someone tell me what answer you get and i'll see if it matches what i have in my text book. If it does i'd be interested to know exactly how you entered it into the calculator.

If the above isnt clear it's the sq root of 62.9∠88.2 / 0.00165∠72.3.

Thanks.

2. Jun 6, 2015

### Svein

The nominator should be the square root of 62.9 at an angle of half of 88.2°. Now divide the magnitudes and subtract the angles.

3. Jun 6, 2015

### Jeff Rosenbury

It is still not clear.

If I remember my orders of precedence you mean sqrt(62.9on the angle 88.2) all ÷0.00165on the angle72.3.

Since the square root is set of functions on a number which when the result is squared return the number, what resultant numbers will do that?

Do you know how to multiply and divide in phasor notation?

4. Jun 6, 2015

### Gremlin

I want to take the sq root of 62.9∠88.2 / 0.00165∠72.3

When i do that division on the calculator in isolation i get 38121.2∠15.9

Now if on the calculator (Casio fx-3650P) i do: √38121.2∠15.9 i get 195.2∠15.9

The answer my text book says i should get is 195∠8

5. Jun 6, 2015

### Jeff Rosenbury

That is the most popular answer. It is like saying the square root of 4 is 2. Yet -2 is also an sometimes answer depending on how the question is phrased. There are other phase angles which when doubled add up to 88.2. Which are valid answers depend on the original problem. If the original question is, what is the square root of X, you are of course correct.

But if the original question results from some physical system other, potentially nonsensical, answers may occur.

6. Jun 6, 2015

### momoko

You need to halve the argument so your textbook's answer looks right.

7. Jun 6, 2015

### Gremlin

I did't know that. Is there a way of inputting it into the calculator so the calculator gets it right?

8. Jun 6, 2015

### Svein

Now I see a problem. Is the square root over the complete expression or just over the nominator?

9. Jun 6, 2015

### vela

Staff Emeritus
Your problem is similar to how you're writing. The question people have been asking you is whether you're trying to calculate
$$\sqrt{\frac{62.9 e^{i88.2^\circ}}{0.00165 e^{i72.3^\circ}}}$$ or
$$\frac{\sqrt{62.9 e^{i88.2^\circ}}}{0.00165 e^{i72.3^\circ}}.$$ If you simply used parentheses, it would have cleared up the ambiguity, i.e., √ (62.9∠88.2 / 0.00165∠72.3). Therein lies the root of your problem. Use parentheses as appropriate on your calculator as well.

10. Jun 6, 2015

### Gremlin

When i input it into the calculator exactly as you have it above i get "math error".

11. Jun 6, 2015

### SammyS

Staff Emeritus
√ ((62.9∠88.2) / (0.00165∠72.3))

12. Jun 6, 2015

### Staff: Mentor

The square root of 4 is 2. Period.
Not if the question is, "what is the square root of 4?" The expression $\sqrt{4}$ has only one value, +2. It is true that 4 has two square roots, but by definition and long usage, the symbol $\sqrt{x}$ means the positive square root of x. I am assuming here the real-valued function, where x $\ge$ 0.

13. Jun 6, 2015

### Jeff Rosenbury

Did I claim differently?

I claimed that if the question is phrased differently, the other root needs to be considered. Since the OP clearly wasn't presenting the problem in the way it was originally phrased, I thought this point should be mentioned.

14. Jun 7, 2015

### Staff: Mentor

Yes. Here is what you said, verbatim:
-2 is never the answer if the question is, "what is the square root of 4?"

However, if the question is, "what are the two numbers whose square is 4?", then I agree that the two numbers are 2 and -2. If we're talking about the square root of a number, we're talking about the principal, or positive square root.

15. Jun 7, 2015

### Gremlin

Math error.

16. Jun 7, 2015

### Svein

Your calculator cannot deal with this notation. Do it this way:
1. Divide 62.9 by 0.00165 (= 38 121.21212)
2. Subtract 72.3 from 88.2 (= 15.9)
3. Take the square root of the result in 1 (= 195.246). That is the magnitude of the answer.
4. Divide the result in 3. by two (= 7.95). That is the angle of the answer.
There you are.

17. Jun 7, 2015

### Gremlin

So if you don't know to sq root the result of dividing the magnitudes and then half the result of subtracting the angles (which i didn't or had forgotten), you'd be knackered basically.

Thanks for clearing that up.

18. Jun 7, 2015

### Jeff Rosenbury

So what you are saying is: "What is the square root of 4?" brings an answer of 2 (i.e. the most popular answer; other unpopular answers being wrong.) But if phrased as "What are two numbers whose square is 4?", then 2 and -2 are both correct?

More particularly if phrased as "In a 60 Hz system, what voltages (across a purely resistive load) would give a power 195 on an angle 8º?" Then, other answers might matter.

A deeper understanding of the math than simply knowing how to push buttons on the calculator is essential to good engineering. I would think it helps in science as well.

19. Jun 7, 2015

### Svein

Oh, I agree completely. But introducing mathematical rigor on a too early stage is likely to confuse.
I could, of course have added: "15.9° under the root sign is also equal to 360° + 15.9°, therefore halving the angle gives two answers; 7.95° and 187.95°". But, as his textbook states that the answer is 8°, I let well enough alone.

20. Jun 7, 2015

### vela

Staff Emeritus
Generally speaking, it's not a good idea to rely on a calculator to find answers you don't know how, in principle, to get by hand.

21. Jun 7, 2015

### vela

Staff Emeritus
The phrase "the square root of 4" refers, by convention, to the principal square root of 4, denoted by $\sqrt 4$, which, by definition, is always +2 and never -2. Is -2 a square root of 4? Sure, but it's not an answer to the question, "What is the square root of 4?"

But this is a completely different question (and an ill-posed one as there are no units on the number 195). No one is arguing that if you want to solve the equation $x^2=4$ that you need to include the second root to be complete, but that's different than saying $x = \sqrt{4}$.

Of course it is, and no one is saying it isn't. At the same time, it's important that one can communicate clearly and follow standard conventions. You don't write, for example, the quadratic equation as
$$x = \frac{-b + \sqrt{b^2-4ac}}{2a}$$ and then claim that everyone should know there are two square roots so using the plus/minus symbol is unnecessary or incorrect.

22. Jun 7, 2015

### Ray Vickson

I must disagree: many descriptions distinguish between "square root of $x$" and the "sqrt(x)" or "$\sqrt{x}$ or "$x^{1/2}$" function. According to http://en.wikipedia.org/wiki/Square_root : "In mathematics, a square root of a number a is a number y such that y^2 = a, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a.[1] For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16." It goes on to say further: "Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a^(1/2)." My understanding of the terms has for decades been in line with the Wikipedia explanation.

23. Jun 7, 2015

### Staff: Mentor

Ray, my remarks refer primarily to what Jeff Rosenbury said. I took what he said as implying that for $\sqrt{4}$, 2 was the "most popular" answer, but that there was another answer, -2. vela seems to be saying pretty much the same as what I said.

24. Jun 7, 2015

### Ray Vickson

I agree: he said "the" square root, rather than "a" square root (but he did not use the symbol $\sqrt{4}$).

25. Jun 7, 2015

### Jeff Rosenbury

I met a grad student when I was in school. He told me a story of how, when a general at his job had said something really stupid, he insisted the general was wrong. Thus instead of being an employed engineer he was in grad school after being fired when the general canceled his project rather than be embarrassed.

Since then I've been more open to other people's opinions even when I think they are stupid. I've learned to ask what people mean instead of simply assuming they are idiots. I use less precise language at the beginning of conversations to help us as a group to arrive at the right answer rather than degenerate into acrimony. Sometimes it even happens that they are right and I'm wrong.

Thus I tend to choose words like "unpopular" instead of "what kind of idiot would think that?"

The goal is to reach the truth, not score points by "proving" the other guy is wrong.

I have not disagreed with your points. So why is this discussion still going on?