Embarrassed to ask

07162014, 01:48 AM
Post: #1




Embarrassed to ask
Why is 8^(2/3)=4 instead of 4?


07162014, 02:08 AM
Post: #2




RE: Embarrassed to ask  
07162014, 03:48 AM
Post: #3




RE: Embarrassed to ask
(07162014 02:08 AM)Waon Shinyoe Wrote: 8^(2/3)=(8^(2/3))=4 FWIW, I explain that to my students as an "algebraic order of operations" thing. To evaluate 2², the exponent must be handled before the negative sign. So 2² means (2²), not (2)². <0ɸ0> Joe 

07162014, 06:01 AM
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RE: Embarrassed to ask
(07162014 02:08 AM)Waon Shinyoe Wrote: (8)^(2/3)=4 Careful  fractional exponentiation of negative numbers is a thorny issue, and the answer is different depending on the assumed domain and the precise definition of the exponentiation operator (which was not specified here). Although 4 is typically the accepted answer in the real domain, a more general interpretation (based on analytic continuation of exponentiation in the complex domain) is that there are THREE answers of the form \[(8)^{2/3}= \{4\omega, 4\omega^2, 4\},\] where \[\omega = \frac{1}{2} + i\frac{\sqrt{3}}{2}\] is the socalled principal cube root of unity. (Try evaluating \(\omega^3\) if you don't believe it.) The first of these three answers, \[4\omega = 2 + i2\sqrt{3},\] would generally be considered the principal answer. If you don't believe me, consult a higher authority: try evaluating (8)^(2/3) on an HP42S and see what it says. John 

07162014, 06:53 AM
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RE: Embarrassed to ask  
07162014, 02:02 PM
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RE: Embarrassed to ask
(07162014 01:48 AM)lrdheat Wrote: Why is 8^(2/3)=4 instead of 4?It is a convention to have the unary minus as a shortcut for (your example) 08^(2/3). So, a written negative number is not atomic, nor does the unary minus have a priority different from the binary minus. This is strange enough to have many people fall for expectations like yours. 

07162014, 02:35 PM
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RE: Embarrassed to ask
(07162014 02:02 PM)Thomas Radtke Wrote:(07162014 01:48 AM)lrdheat Wrote: Why is 8^(2/3)=4 instead of 4?It is a convention to have the unary minus as a shortcut for (your example) 08^(2/3). Hmmh, AFAIK the convention is the unary minus corresponds to a multiplication times (1). So, in this case, 8^(2/3) = (1)*8^(2/3) = (1)*4 = 4. d:) 

07162014, 02:53 PM
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RE: Embarrassed to ask
Thanks all...(order of precedence on calculating device).. I thought I was losing it!


07162014, 05:07 PM
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RE: Embarrassed to ask  
07162014, 05:11 PM
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RE: Embarrassed to ask  
07162014, 07:48 PM
(This post was last modified: 07162014 07:53 PM by jebem.)
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RE: Embarrassed to ask
(07162014 06:01 AM)John R Wrote:(07162014 02:08 AM)Waon Shinyoe Wrote: (8)^(2/3)=4 Interesting... and my HP15C gives me an Domain Error. I mean, using 8^( 2/3) Jose Mesquita RadioMuseum.org member 

07162014, 08:38 PM
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RE: Embarrassed to ask
(07162014 07:48 PM)jebem Wrote: Interesting... and my HP15C gives me an Domain Error. Try setting the 15C to complex mode (g SF 8). Ceci n'est pas une signature. 

07162014, 08:57 PM
(This post was last modified: 07162014 09:07 PM by jebem.)
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RE: Embarrassed to ask
(07162014 08:38 PM)Mark Hardman Wrote:(07162014 07:48 PM)jebem Wrote: Interesting... and my HP15C gives me an Domain Error. Yes, the HP15C answer is a complex number: 2.0000 for real and 3,4641 for imaginary part. where 3,4641 should be the approximation for 2*sqrt(3) and of course my HP42S doesn't even need to be told to enter complex mode, as it just gave a straight answer in the x stack register: x: 2.0000 i3.4641 And just for the record, a HP27S answer for 8^( 2/3) is: ERROR: NEG^NONINTEGER And Google itself is answering like this: (8)^(2 / 3) = 2 + 3,46410162 i Jose Mesquita RadioMuseum.org member 

07162014, 09:08 PM
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RE: Embarrassed to ask  
07162014, 09:43 PM
(This post was last modified: 07162014 09:48 PM by John R.)
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RE: Embarrassed to ask
(07162014 08:57 PM)jebem Wrote: and of course my HP42S doesn't even need to be told to enter complex mode, as it just gave a straight answer in the x stack register: If desired  possibly for strict compatibility with the 41C series  the 42S can be told to suppress complex results that would otherwise be generated from real arguments. This can be done by selecting RRES in the MODES menu, which sets flag 74. John 

07172014, 06:45 AM
Post: #16




RE: Embarrassed to ask
(07162014 09:43 PM)John R Wrote:(07162014 08:57 PM)jebem Wrote: and of course my HP42S doesn't even need to be told to enter complex mode, as it just gave a straight answer in the x stack register: Thanks, John! Jose Mesquita RadioMuseum.org member 

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