## Can't understand this result from a few calculators...

Discussion around the Swiss Micros DM42 calculator.
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### Can't understand this result from a few calculators...

Hello everyone,

I cannot understand this result from a few calculators. Please could somebody explain what is happening.

When I was in High School (more than 35 years ago), I was taught that a decimal power of a negative number is a complex number. Let’s consider (-3)^3.6 and look at the result given by a few SCIENTIFIC calculators.

Result-1: Math Error.
This result is given by Casio FX82 (very old calculator, bought in 1984), Casio FX 4500PA programmable, Casio FX100AU, HP10s+ and a few more calculators that I tested it on (I see a lot of calculators in my work). These are low end calculators that have no capacity to deal with complex numbers and therefore are indicating that the calculation is beyond their scope. Free42 says “Invalid data” when in “realres” mode, and HP Prime says “undef” also in “real” CAS mode.

Result-2: 52.19591521
This result is given by Casio FX82AU (the newish version of the old FX82), and HP300s. While these are of cheaper variety, I was shocked to see this result on Casio Classpad CP400 when in “Real” mode. Interestingly, all these calculators have the ability to display fractions (book view I think they call it). It appears that they are substituting (18/5) for 3.6 and then working out the fifth root of (-3)^18. As (-3)^18 is positive, the fifth root is a real number. But the point is, why should the calculators overthink and make this substitution in the first place? I didn’t ask for it.

Result-3: 16.1294-49.6413i
Given by HP48gII and HP50g. When in “complex” mode Classpad CP400, HP42s, Free42 (and therefore DM42) give this result. Mathematica gives this answer. WolframAlpha says this is the principal root of (-3)^3.6. Wolf then proceeds to give all five roots of (-3)^(18/5) if one decides to make the substitution of (18/5) for 3.6. One of the roots is Result-2 above. But only if 3.6 is replaced by (18/5).

From what I learnt Result-3 is the correct one. But what is happening here? Are some calculators, including Classpad when in “real mode”, giving wrong answers? These many? Or is it that my understanding of numbers is wrong?

Regards,

HPMike
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### Re: Can't understand this result from a few calculators...

It looks like Result 2 is the real part of the complex number in polar coordinates. That would correspond to the vector magnitude, and the imaginary part not shown is the phase angle.
DM15L, S/N 00548. DM42, SN: 00159.

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### Re: Can't understand this result from a few calculators...

HPMike wrote:
Thu Mar 01, 2018 10:58 pm
It looks like Result 2 is the real part of the complex number in polar coordinates. That would correspond to the vector magnitude, and the imaginary part not shown is the phase angle.

Result-2 is the modulus (distance from the origin to the vector in the complex plane) of the complex number. As you said correctly the argument of the complex number (which is the angle in the complex plane) is not shown by the calculators. [16.1294-49.6413i (in rectangular coordinates) is equivalent to a modulus of 52.19591521 and an argument of -72 degrees]. As the result of (-3)^3.6 is two dimensional, i.e., comprising both a real part and an imaginary part, I feel it is wrong to show only the modulus and ignore the argument. Some, including Classpad, are making this mistake. The reason for my being so particular about the Classpad is that it is used by all Year 11 and Year 12 Mathematics students here in Western Australia. The students would not know if the calculator is giving them a wrong answer.

I am not sure if I am wrongly saying the calculators are wrong or they are really wrong.

ijabbott
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### Re: Can't understand this result from a few calculators...

HPMike wrote:
Thu Mar 01, 2018 10:58 pm
It looks like Result 2 is the real part of the complex number in polar coordinates. That would correspond to the vector magnitude, and the imaginary part not shown is the phase angle.
Is it really showing the magnitude (modulus) and not just happening to show the one result that lies on the real axis (when the calculator is in "Real" mode)? I wonder what result the Classpad (in "Real" mode) would show for a negative base raised to an irrational exponent (or as close to irrational as you can get on a fixed precision calculator)? E.g. what would it show for (-3)^pi?

Testing-42
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### Re: Can't understand this result from a few calculators...

HPMike wrote:
Thu Mar 01, 2018 10:58 pm
It looks like Result 2 is the real part of the complex number in polar coordinates. That would correspond to the vector magnitude, and the imaginary part not shown is the phase angle.
Damn, it is a pity those calculators do not give back a partial rectangular coordinate. That way the result would be "really correct". Posts: 12
Joined: Fri Nov 17, 2017 9:53 am
Location: Perth, Australia

### Re: Can't understand this result from a few calculators...

ijabbott wrote:
Fri Mar 02, 2018 5:20 pm
HPMike wrote:
Thu Mar 01, 2018 10:58 pm
It looks like Result 2 is the real part of the complex number in polar coordinates. That would correspond to the vector magnitude, and the imaginary part not shown is the phase angle.
Is it really showing the magnitude (modulus) and not just happening to show the one result that lies on the real axis (when the calculator is in "Real" mode)? I wonder what result the Classpad (in "Real" mode) would show for a negative base raised to an irrational exponent (or as close to irrational as you can get on a fixed precision calculator)? E.g. what would it show for (-3)^pi?
To answer your question, Classpad (in "Real" mode) says "Non-Real in Calc", for (-3)^pi (I have checked it again now). It says the same for all irrational powers for a negative number and also for (-3)^0.5. Let me clarify.

From what I can figure out, Classpad (in "Real" mode) gives a real number answer to a negative number raised to a decimal power when
(a) the decimal power can be replaced with a fraction and
(b) the numerator of the fraction is an even number.

If the above two conditions are not satisfied, Classpad returns "Non-Real in Calc".

For example, the answer obtained is 54.13094523 for (-5)^2.48. In this, 2.48 could be replaced by 62/25 and the numerator is an even number. The calculator then, it appears, works out the 25th root of (-5)^62. However, if you attempt (-5)^2.58, you will get "Non-Real in Calc". The power 2.58 can be replaced by 129/50. As the numerator is an odd number (-5)^129 is negative and the 50th root is a complex number.

ijabbott
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### Re: Can't understand this result from a few calculators...

Sat Mar 03, 2018 12:32 am
From what I can figure out, Classpad (in "Real" mode) gives a real number answer to a negative number raised to a decimal power when
(a) the decimal power can be replaced with a fraction and
(b) the numerator of the fraction is an even number.

If the above two conditions are not satisfied, Classpad returns "Non-Real in Calc".

For example, the answer obtained is 54.13094523 for (-5)^2.48. In this, 2.48 could be replaced by 62/25 and the numerator is an even number. The calculator then, it appears, works out the 25th root of (-5)^62. However, if you attempt (-5)^2.58, you will get "Non-Real in Calc". The power 2.58 can be replaced by 129/50. As the numerator is an odd number (-5)^129 is negative and the 50th root is a complex number.

Yes, that makes sense (certainly, it makes more sense than just returning the absolute value of the complex result!). I just tried it on my FX-CG50 PRIZM and it behaves the same as above (not that surprising, really).

EDIT

Actually, I think condition (b) is:

(b) the denominator of the (simplified) fraction is an odd number.

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### Re: Can't understand this result from a few calculators...

ijabbott wrote:
Sat Mar 03, 2018 2:02 am

Yes, that makes sense (certainly, it makes more sense than just returning the absolute value of the complex result!). I just tried it on my FX-CG50 PRIZM and it behaves the same as above (not that surprising, really).

EDIT

Actually, I think condition (b) is:

(b) the denominator of the (simplified) fraction is an odd number.
Regarding condition (b), I accept. If the denominator is an odd number, then one can do roots of both positive and negative numbers. Thanks for the correction. I was thinking more in the lines that if the numerator is an even number then the exponentiation makes the result positive and then any root of that number is possible.

However, what I would like to know is whether substituting (62/25 ) for 2.48 is correct. I feel the calculator should not substitute just because it can.

Walter
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