Imaginary exponents
Main article: Euler's formula
See also: Complex exponents with a positive real base
In this animation N takes various increasing values from 1 to 100. The computation of (1 +
iπ
/
N
)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 +
iπ
/
N
)N. It can be seen that as N gets larger (1 +
iπ
/
N
)N approaches a limit of −1.
Fundamentally, Euler's identity asserts that {\displaystyle e^{i\pi }} e^{{i\pi }} is equal to −1. The expression {\displaystyle e^{i\pi }} e^{{i\pi }} is a special case of the expression {\displaystyle e^{z}} e^{z}, where z is any complex number. In general, {\displaystyle e^{z}} e^{z} is defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition is:
{\displaystyle e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.} {\displaystyle e^{z}=\lim _{n\to \infty }\left(1+{\frac {z}{n}}\right)^{n}.}
Euler's identity therefore states that the limit, as n approaches infinity, of {\displaystyle (1+i\pi /n)^{n}} {\displaystyle (1+i\pi /n)^{n}} is equal to −1. This limit is illustrated in the animation to the right.
Euler's formula for a general angle
Euler's identity is a special case of Euler's formula, which states that for any real number x,
{\displaystyle e^{ix}=\cos x+i\sin x} {\displaystyle e^{ix}=\cos x+i\sin x}
where the inputs of the trigonometric functions sine and cosine are given in radians.
In particular, when x = π,
{\displaystyle e^{i\pi }=\cos \pi +i\sin \pi .} {\displaystyle e^{i\pi }=\cos \pi +i\sin \pi .}
Since
{\displaystyle \cos \pi =-1} {\displaystyle \cos \pi =-1}
and
{\displaystyle \sin \pi =0,} \sin \pi =0,
it follows that
{\displaystyle e^{i\pi }=-1+0i,} e^{i\pi }=-1+0i,
which yields Euler's identity:
{\displaystyle e^{i\pi }+1=0.} e^{i\pi }+1=0.
Geometric interpretation
Any complex number {\displaystyle z=x+iy} z=x+iy can be represented by the point {\displaystyle (x,y)} (x,y) on the complex plane. This point can also be represented in polar coordinates as {\displaystyle (r,\theta )} (r,\theta ), where r is the absolute value of z (distance from the origin), and {\displaystyle \theta } \theta is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of {\displaystyle (r\cos \theta ,r\sin \theta )} {\displaystyle (r\cos \theta ,r\sin \theta )}, implying that {\displaystyle z=r(\cos \theta +i\sin \theta )} {\displaystyle z=r(\cos \theta +i\sin \theta )}. According to Euler's formula, this is equivalent to saying {\displaystyle z=re^{i\theta }} {\displaystyle z=re^{i\theta }}.
Euler's identity says that {\displaystyle -1=e^{i\pi }} -1=e^{{i\pi }}. Since {\displaystyle e^{i\pi }} e^{{i\pi }} is {\displaystyle re^{i\theta }} {\displaystyle re^{i\theta }} for r = 1 and {\displaystyle \theta =\pi } \theta =\pi , this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is {\displaystyle \pi } \pi radians.
Additionally, when any complex number z is multiplied by {\displaystyle e^{i\theta }} e^{i\theta }, it has the effect of rotating z counterclockwise by an angle of {\displaystyle \theta } \theta on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point {\displaystyle \pi } \pi radians around the origin has the same effect as reflecting the point across the origin.
Generalizations
Euler's identity is also a special case of the more general identity that the nth roots of unity, for n > 1, add up to 0:
{\displaystyle \sum _{k=0}^{n-1}e^{2\pi i{\frac {k}{n}}}=0.} {\displaystyle \sum _{k=0}^{n-1}e^{2\pi i{\frac {k}{n}}}=0.}
Euler's identity is the case where n = 2.
In another field of mathematics, by using quaternion exponentiation, one can show that a similar identity also applies to quaternions. Let {i, j, k} be the basis elements; then,
{\displaystyle e^{{\frac {1}{\sqrt {3}}}(i\pm j\pm k)\pi }+1=0.} {\displaystyle e^{{\frac {1}{\sqrt {3}}}(i\pm j\pm k)\pi }+1=0.}
In general, given real a1, a2, and a3 such that a12 + a22 + a32 = 1, then,
{\displaystyle e^{\left(a_{1}i+a_{2}j+a_{3}k\right)\pi }+1=0.} {\displaystyle e^{\left(a_{1}i+a_{2}j+a_{3}k\right)\pi }+1=0.}
For octonions, with real an such that a12 + a22 + ... + a72 = 1, and with the octonion basis elements {i1, i2, ..., i7},
{\displaystyle e^{\left(a_{1}i_{1}+a_{2}i_{2}+\dots +a_{7}i_{7}\right)\pi }+1=0.} {\displaystyle e^{\left(a_{1}i_{1}+a_{2}i_{2}+\dots +a_{7}i_{7}\right)\pi }+1=0.}
History
It has been claimed that Euler's identity appears in his monumental work of mathematical analysis published in 1748, Introductio in analysin infinitorum.[15] However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it.[16] Moreover, while Euler did write in the Introductio about what we today call Euler's formula,[17] which relates e with cosine and sine terms in the field of complex numbers, the English mathematician Roger Cotes (who died in 1716, when Euler was only 9 years old) also knew of this formula and Euler may have acquired the knowledge through his Swiss compatriot Johann Bernoulli.[16]
Robin Wilson states the following.[18]
We've seen how it [Euler's identity] can easily be deduced from results of Johann Bernoulli and Roger Cotes, but that neither of them seem to have done so. Even Euler does not seem to have written it down explicitly – and certainly it doesn't appear in any of his publications – though he must surely have realized that it follows immediately from his identity [i.e. Euler's formula], eix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly….
See also
De Moivre's formula
Exponential function
Gelfond's constant
Notes
The term "Euler's identity" (or "Euler identity") is also used elsewhere to refer to other concepts, including the related general formula eix = cos x + i sin x,[1] and the Euler product formula.[2]
References
Dunham, 1999, p. xxiv.
Stepanov, S. A. (7 February 2011). "Euler identity". Encyclopedia of Mathematics. Retrieved 7 September 2018.
Gallagher, James (13 February 2014). "Mathematics: Why the brain sees maths as beauty". BBC News Online. Retrieved 26 December 2017.
Paulos, 1992, p. 117.
Nahin, 2006, p. 1.
Nahin, 2006, p. xxxii.
Reid, chapter e.
Maor, p. 160, and Kasner & Newman, p. 103–104.
Wells, 1990.
Crease, 2004.
Zeki et al., 2014.
Nahin, Paul (2011). Dr. Euler's fabulous formula : cures many mathematical ills. Princeton University Press. ISBN 978-0691118222.
Stipp, David (2017). A most elegant equation : Euler's formula and the beauty of mathematics (First ed.). Basic Books. ISBN 978-0465093779.
Wilson, Robin (2018). Euler's pioneering equation : the most beautiful theorem in mathematics. Oxford: Oxford University Press. ISBN 978-0198794936.
Conway & Guy, p. 254–255.
Sandifer, p. 4.
Euler, p. 147.
Wilson, p. 151-152.
Sources
Conway, John H., and Guy, Richard K. (1996), The Book of Numbers, Springer ISBN 978-0-387-97993-9
Crease, Robert P. (10 May 2004), "The greatest equations ever", Physics World [registration required]
Dunham, William (1999), Euler: The Master of Us All, Mathematical Association of America ISBN 978-0-88385-328-3
Euler, Leonhard (1922), Leonhardi Euleri opera omnia. 1, Opera mathematica. Volumen VIII, Leonhardi Euleri introductio in analysin infinitorum. Tomus primus, Leipzig: B. G. Teubneri
Kasner, E., and Newman, J. (1940), Mathematics and the Imagination, Simon & Schuster
Maor, Eli (1998), e: The Story of a number, Princeton University Press ISBN 0-691-05854-7
Nahin, Paul J. (2006), Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, Princeton University Press ISBN 978-0-691-11822-2
Paulos, John Allen (1992), Beyond Numeracy: An Uncommon Dictionary of Mathematics, Penguin Books ISBN 0-14-014574-5
Reid, Constance (various editions), From Zero to Infinity, Mathematical Association of America
Sandifer, C. Edward (2007), Euler's Greatest Hits, Mathematical Association of America ISBN 978-0-88385-563-8
Stipp, David (2017), A Most Elegant Equation: Euler's formula and the beauty of mathematics, Basic Books
Wells, David (1990). "Are these the most beautiful?". The Mathematical Intelligencer. 12 (3): 37–41. doi:10.1007/BF03024015.
Wilson, Robin (2018), Euler's Pioneering Equation: The most beautiful theorem in mathematics, Oxford University Press
Zeki, S.; Romaya, J. P.; Benincasa, D. M. T.; Atiyah, M. F. (2014), "The experience of mathematical beauty and its neural correlates", Frontiers in Human Neuroscience, 8: 68, doi:10.3389/fnhum.2014.00068, PMC 3923150, PMID 24592230
External links
Wikiquote has quotations related to: Euler's identity
Complete derivation of Euler's identity
Intuitive understanding of Euler's formula
Categories: ExponentialsMathematical identitiesE (mathematical constant)Theorems in complex analysisLeonhard Euler
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MANAGING ORGANIZATIONS
4 Things to Consider Before You Start Using AI in Personnel Decisions
by Peter Cappelli
November 03, 2020
Summary Save Share Print 8.95 Buy Copies
Juan Moyano/Stocksy
The initial promise of artificial intelligence as a broad-based tool for solving business problems has given way to something much more limited but still quite useful: algorithms from data science that make predictions better than we have been able to do so far.
In contrast to standard statistical models that focus on one or two factors already known to be associated with an outcome like job performance, machine-learning algorithms are agnostic about which variables have worked before or why they work. The more the merrier: It throws them all together and produces one model to predict some outcome like who will be a good hire, giving each applicant a single, easy-to-interpret score as to how likely it is that they will perform well in a job.
INSIGHT CENTER
AI and Equality
Designing systems that are fair for all.
No doubt because the promise of these algorithms was so great, the recognition of their limitations has also gotten a lot of attention, especially given the fact that if the initial data used to build the model is biased, then the algorithm generated from that data will perpetuate that bias. The best-known examples have been in organizations that discriminated against women in the past where job performance data is also biased, and that means algorithms based on that data will also be biased.
So how should employers proceed as they contemplate adopting AI to make personnel decisions? Here are four considerations:
The algorithm may be less biased than the existing practices that generate the data in the first place. Let’s not romanticize how poor human judgment is and how disorganized most of our people management practices are now. When we delegate hiring to individual supervisors, for example, it is quite likely that they may each have lots of biases in favor of and against candidates based on attributes that have nothing to do with good performance: Supervisor A may favor candidates who graduated from a particular college because she went there, while Supervisor B may do the reverse because he had a bad experience with some of its graduates. At least algorithms treat everyone with the same attributes equally, albeit not necessarily fairly.
We may not have good measures of all of the outcomes we would like to predict, and we may not know how to weight the various factors in making final decisions. For example, what makes for a “good employee”? They have to accomplish their tasks well, they also should get along with colleagues well, fit in with the “culture,” stay with us and not quit, and so forth. Focusing on just one aspect where we have measures will lead to a hiring algorithm that selects on that one aspect, often when it does not relate closely to other aspects, such as a salesperson who is great with customers but miserable with co-workers.
Here again, it isn’t clear that what we are doing now is any better: An individual supervisor making a promotion decision may be able in theory to consider all those criteria, but each assessment is loaded with bias, and the way they are weighted is arbitrary. We know from rigorous research that the more hiring managers use their own judgment in these matters, the worse their decisions are.
The data that AI uses may raise moral issues. Algorithms that predict turnover, for example, now often rely on data from social media sites, such as Facebook postings. We may decide that it is an invasion of privacy to gather such data about our employees, but not using it comes at the price of models that will predict less well.
It may also be the case that an algorithm does a good job overall in predicting something for the average employee but does a poor job for some subset of employees. It might not be surprising, for example, to find that the hiring models that pick new salespeople do not work well at picking engineers. Simply having separate models for each would seem to be the solution. But what if the different groups are men and women or whites and African Americans, as appears to be the case? In those cases, legal constraints prevent us from using different practices and different hiring models for different demographic groups.
It is often hard, if not impossible, to explain and justify the criteria behind algorithmic decisions. In most workplaces now, we at least have some accepted criteria for making employment decisions: He got the opportunity because he has been here longer; she was off this weekend because she had that shift last weekend; this is the way we have treated people before. If I don’t get the promotion or the shift I want, I can complain to the person who made the decision. He or she has a chance to explain the criterion and may even help me out next time around if the decision did not seem perfectly fair.
When we use algorithms to drive those decisions, we lose the ability to explain to employees how those decisions were made. The algorithm simply pulls together all the available information to construct extremely complicated models that predict past outcomes. It would be highly unlikely if those outcomes corresponded to any principle that we could observe or explain other than to say, “The overall model says this will work best.” The supervisor can’t help explain or address fairness concerns.
Especially where such models do not perform much better than what we are already doing, it is worth asking whether the irritation they will cause employees is worth the benefit. The advantage, say, of just letting the most senior employee get first choice in picking his or her schedule is that this criterion is easily understood, it corresponds with at least some accepted notions of fairness, it is simple to apply, and it may have some longer-term benefits, such as increasing the rewards for sticking around. There may be some point where algorithms will be able to factor in issues like this, but we are nowhere close to that now.
Algorithmic models are arguably no worse than what we are doing now. But their fairness problems are easier to spot because they happen at scale. The way to solve them is to get more and better measures — data that is not biased. Doing that would help even if we were not using machine-learning algorithms to make personnel decisions.
Peter Cappelli is the George W. Taylor Professor of Management at the Wharton School and a director of its Center for Human Resources. He is the author of several books, including Will College Pay Off? A Guide to the Most Important Financial Decision You’ll Ever Make (PublicAffairs, 2015).
https://www.linkedin.com/in/jesse-west-james-7175b9179/
Jesse James AKA PANDA
Peace Maker International Unilateral Trade Business ecosystem
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