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FactChecker
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IMO, in complex analysis, ##z^{1/2}## is not necessarily assumed to mean the principal square root.Mark44 said:
Regarding i^2, and why it's -1 and not 1
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In summary, the use of ##\sqrt{-1}## as a solution to the equation ##x^2 + 1 = 0## is arbitrary and either the positive or negative root can be used. The convention is to use the positive root, but this does not have any mathematical significance. In complex numbers, the symbol ##\sqrt w## represents the principal square root function, which has a single value and is not equivalent to using the ##\pm## notation. The use of ##i## in special relativity is not a helpful convention and can be dropped without affecting the validity of the theory. The sign convention in the line element in relativity is simply a matter of convenience and personal
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Erik 05
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Mark44 said:
##(-4)^{1/2}=\pm\sqrt{-4}=\pm 2i##
##\sqrt 9 = +(9^{1/2})##, which is 3
I don't believe I'm misusing or confusing anything.
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Mark44
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Although there are two square roots of -4, I believe the convention is to evaluate ##(-4)^{1/2}## to its principal value, which is 2i.Erik 05 said:
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Erik 05
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Well, that's the convention I use, as I said, less confusing, but at the end of the day it might well come down to whether you are doing maths or physics, or whatever problem you are solving - often you are only interested in positive values, or real numbers, though not always.Mark44 said:
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You mean you could do an experiment? Fire a beam of numbers through a square root machine and see whether they come out as one beam of principal square roots, or (like Stern-Gerlach) two beams, each corresponding to a separate root?Erik 05 said:
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Erik 05
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Not sure, would the numbers be entangled? Maybe some quantum computing algorithm...
Ok, I did a physics degree a few years ago, having a background in maths, and I have to say that some of the maths that physicists do did strike me as, well, heresy, almost.
Ok, I did a physics degree a few years ago, having a background in maths, and I have to say that some of the maths that physicists do did strike me as, well, heresy, almost.
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This thread has become far more than I had hoped. I've learned a lot more about complex numbers than I anticipated I would. It's also interesting to see the limits of convention and how that messes with people.
Can we have a similar thread with the obelus, and grouping conventions (like mulitplication by juxtaposition)?
8÷2(2 + 2) is the latest trendy one, I believe.
(I kid, I kid)
Can we have a similar thread with the obelus, and grouping conventions (like mulitplication by juxtaposition)?
8÷2(2 + 2) is the latest trendy one, I believe.(I kid, I kid)
- #43
Svein
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Which is the point I was trying to make above, but forgot to stress: The concept of "principal root" is meaningless in complex analysis. A (possibly) better example: What is the square root of i? Skipping the small steps of adding multiples of 2πi, we have [itex]i=e^{\frac{i \pi}{2}+2n\pi i} [/itex] and therefore [itex]\sqrt{i}=e^{\frac{i\pi}{4}+n\pi i}=(\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2})\cdot (-1)^{n} [/itex].FactChecker said:
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I wouldn't say that. The principle root is not assumed, but it is a fundamental concept that is studied thoroughly in complex analysis.Svein said:
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Why do you need the ##(-1)^n## if ##\sqrt 2 = p(2)(-1)^n##? Where ##p(2)## is the principal square root of ##2##?Svein said:
Also, if ##\sqrt 2 = \pm p(2)##, then you now have a problem with the expression ##\frac{\sqrt{2}}{2}+i\frac{\sqrt{2}}{2}##, as this is now a set of four complex numbers, rather than one or two.
Like it or not, you are forced to use the concept of ##\sqrt 2## being a single number - or invent a new notation, such as ##p(2)## to force it to take a single value.
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jedishrfu
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It seems we have exhaustively explored all the various viewpoints of this topic and all beginning to reiterate key areas of thought and with that in mind it’s good to tank everyone for contributing and to close this thread now.
Closing now with an excellent summary from @pbuk :
When we are working with complex numbers there are in general 2 solutions to the equation ## w^2 = z ##.
In some branches and applications of mathematics we don't need to distinguish between the solutions, and so in these contexts when we write ## w = \sqrt z ## we mean that ## w ## is any number that satisfies the equation ## w^2 = z ##, for example ## w = \sqrt{-4} \implies w \in \{2i, -2i\} ##.
In other branches and applications of mathemetics it is useful to define the 'principal square root' such that when we write ## w = \sqrt z ## we mean that ## w ## is the principal square root of ## z ##; for example ## w = \sqrt{-4} \implies w = 2i ##.
When we represent complex numbers as points in the Argand Plane, we define the principal n'th root of a number ## \sqrt[n]{z} ## as the solution ## (r, \theta)^n = z ## with the smallest positive value of ## \theta ##. It is worth noting:
Thank you,
The Mentors
Closing now with an excellent summary from @pbuk :
When we are working with complex numbers there are in general 2 solutions to the equation ## w^2 = z ##.
In some branches and applications of mathematics we don't need to distinguish between the solutions, and so in these contexts when we write ## w = \sqrt z ## we mean that ## w ## is any number that satisfies the equation ## w^2 = z ##, for example ## w = \sqrt{-4} \implies w \in \{2i, -2i\} ##.
In other branches and applications of mathemetics it is useful to define the 'principal square root' such that when we write ## w = \sqrt z ## we mean that ## w ## is the principal square root of ## z ##; for example ## w = \sqrt{-4} \implies w = 2i ##.
When we represent complex numbers as points in the Argand Plane, we define the principal n'th root of a number ## \sqrt[n]{z} ## as the solution ## (r, \theta)^n = z ## with the smallest positive value of ## \theta ##. It is worth noting:
- this is compatiable with the definition that is (almost) always used in ## \mathbb R ## of the principal square root of a positive number where ## x = -2 ## is not a solution to ## x = \sqrt 4 ##.
- computer languages and computer algebra packages often implement a complex-valued
sqrtor similar function which returns the principal square root according to this definition.
Thank you,
The Mentors
<h2>1. What is i^2 and why is it -1?</h2><p>i^2 is the mathematical term for the imaginary number, which is represented by the letter "i". It is defined as the square root of -1. The reason why it is -1 is because when multiplied by itself, it results in -1. This is a fundamental property of imaginary numbers.</p><h2>2. Why is i^2 important in mathematics?</h2><p>i^2 is important in mathematics because it allows us to solve equations that involve negative numbers under the square root. It is also used in various fields of science, such as engineering, physics, and economics.</p><h2>3. How is i^2 different from regular numbers?</h2><p>i^2 is different from regular numbers because it is not a real number. Real numbers are numbers that can be plotted on a number line and include all rational and irrational numbers. Imaginary numbers, on the other hand, cannot be plotted on a number line and include only numbers that involve the square root of -1.</p><h2>4. Can i^2 be simplified?</h2><p>Yes, i^2 can be simplified to -1. This is because when multiplied by itself, it results in -1. However, it is important to note that in some cases, i^2 cannot be simplified further and must be left in its original form.</p><h2>5. How is i^2 used in real-world applications?</h2><p>i^2 is used in various real-world applications, such as in electrical engineering, where it is used to represent the phase difference between two alternating currents. It is also used in signal processing and control systems. In physics, imaginary numbers are used to describe the behavior of quantum particles. In economics, they are used to model complex systems and in finance, they are used to calculate interest rates and exchange rates.</p>
1. What is i^2 and why is it -1?
i^2 is the mathematical term for the imaginary number, which is represented by the letter "i". It is defined as the square root of -1. The reason why it is -1 is because when multiplied by itself, it results in -1. This is a fundamental property of imaginary numbers.
2. Why is i^2 important in mathematics?
i^2 is important in mathematics because it allows us to solve equations that involve negative numbers under the square root. It is also used in various fields of science, such as engineering, physics, and economics.
3. How is i^2 different from regular numbers?
i^2 is different from regular numbers because it is not a real number. Real numbers are numbers that can be plotted on a number line and include all rational and irrational numbers. Imaginary numbers, on the other hand, cannot be plotted on a number line and include only numbers that involve the square root of -1.
4. Can i^2 be simplified?
Yes, i^2 can be simplified to -1. This is because when multiplied by itself, it results in -1. However, it is important to note that in some cases, i^2 cannot be simplified further and must be left in its original form.
5. How is i^2 used in real-world applications?
i^2 is used in various real-world applications, such as in electrical engineering, where it is used to represent the phase difference between two alternating currents. It is also used in signal processing and control systems. In physics, imaginary numbers are used to describe the behavior of quantum particles. In economics, they are used to model complex systems and in finance, they are used to calculate interest rates and exchange rates.
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