Physicist: The beauty of complex numbers (numbers that involve 
) is that the answer to this question is a surprisingly resounding: nopers.
The one thing that needs to be known about is that, by definition, 
. Other than that it behaves like any other number or variable. It turns out that the square root is 
. You can check this the same way that you can check that 2 is the square root of 4: you square it.
![\begin{array}{ll}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^2\\[2mm]=\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)\\[2mm]=\frac{1}{\sqrt{2}}\left(1+i\right)\frac{1}{\sqrt{2}}\left(1+i\right)\\[2mm]=\frac{1}{2}\left(1+i\right)\left(1+i\right)\\[2mm]=\frac{1}{2}\left(1+i+i+i^2\right)\\[2mm]=\frac{1}{2}\left(1+i+i-1\right)\\[2mm]=\frac{1}{2}\left(2i\right)\\=i\end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Bll%7D%5Cleft%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%2B%5Cfrac%7Bi%7D%7B%5Csqrt%7B2%7D%7D%5Cright%29%5E2%5C%5C%5B2mm%5D%3D%5Cleft%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%2B%5Cfrac%7Bi%7D%7B%5Csqrt%7B2%7D%7D%5Cright%29%5Cleft%28%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%2B%5Cfrac%7Bi%7D%7B%5Csqrt%7B2%7D%7D%5Cright%29%5C%5C%5B2mm%5D%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cleft%281%2Bi%5Cright%29%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Cleft%281%2Bi%5Cright%29%5C%5C%5B2mm%5D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%281%2Bi%5Cright%29%5Cleft%281%2Bi%5Cright%29%5C%5C%5B2mm%5D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%281%2Bi%2Bi%2Bi%5E2%5Cright%29%5C%5C%5B2mm%5D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%281%2Bi%2Bi-1%5Cright%29%5C%5C%5B2mm%5D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft%282i%5Cright%29%5C%5C%3Di%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=0 "\begin{array}{ll}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^2\\[2mm]=\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)\\[2mm]=\frac{1}{\sqrt{2}}\left(1+i\right)\frac{1}{\sqrt{2}}\left(1+i\right)\\[2mm]=\frac{1}{2}\left(1+i\right)\left(1+i\right)\\[2mm]=\frac{1}{2}\left(1+i+i+i^2\right)\\[2mm]=\frac{1}{2}\left(1+i+i-1\right)\\[2mm]=\frac{1}{2}\left(2i\right)\\=i\end{array}")
And like any other square root, the negative, 
, is also a solution. So, does have a square root, and it’s not even that hard to find it. No new “super-imaginary” numbers need to be invented.
This isn’t a coincidence. The complex numbers are “algebraically closed“, which means that no matter how weird a polynomial is, it’s roots are always complex numbers. The square roots of , for example, are the solutions of the polynomial 
. So, any cube root, any Nth root, any power, any combination, any whatever of any complex number: still a complex number.
That hasn’t stopped mathematicians from inventing new and terrible number systems. They just didn’t need to in this case.
There is no beauty, no cleverness and no need for complex “numbers”.
Complex “numbers” are not numbers of any kind. They are the result of dysfunctional minds. It is not wrong to define [tex]i^2[/tex] as -1. Definitions are very important. This definition is ill-formed.
If a polynomial of degree n has k pairs of complex roots, then it’s the same as saying the polynomial has (n-k) “real” roots.
Complex numbers are the creation of non-mathematicians. There is no sound mathematics, only nonsense. Complex numbers are not required in Fourier series, they are not required in physics and they not required in any mathematics. But that’s what happens when people with no talent for mathematics (Euler and his followers) are allowed to run away with their misguided ideas.
So, while one can say certain things about a polynomial because of a logically unsound (distorted and ugly) object, that is, the complex number, this does not mean it has a place in mathematics.
Very good posting, short and nevertheless comprising! I would so much appreciate to see upcoming posts on Theory of Numbers, and especially Topology, Manifolds..as these are concepts central to Quantum Mechanics and latest state-of-the-art theories of the Universe etc.
Many thanks for all the wonderful work, may it persevere!
Firstly, there is beauty in Complex numbers because their combinations can give real numbers. Secondly, just because something is ‘imaginary’ doesn’t mean it isn’t worth studying. That goes for a lot of things out of history!
You’re right when you say it wasn’t a mathematician who ‘discovered’ the sq.root of -1, it was an engineer by the name of Rafael Bombelli and Euler who gave the symbol i. What you need to appreciate is the further we syudy physics (QM, SString theory etc) the more we find ourselves going to pure maths.
Perhaps some have not heard of Qaternions where Hamilton was thinking of i and then thought of the spatial combinations. Qaternions are in use today in many fields today.
Many of the ‘greats’ have used i and even if one suggests that complex numbers only provide an alternative to other methods then that has much value in itself. Some might find complex numbers easier to study that other methods.
I just dont understand in your demonstration how you get the 1/2 (from the sq.r of 2 obviously but don’t know how he ended up like this) ?
Would you consider zero a made-up number?
Zero – made up by humans, but referring to a “non created energy” from a “non created realm”. Babylonians used it, Greeks feared it… the concept of Zero is very fundamental but at the same time among the most controversial objects/ notions in history of mankind.
I strongly recommend the reading: Zero: The Biography of a Dangerous Idea: Charles Seife
Worth spending some time over it..
Greetings, and a delightful week to you guys!
“Complex “numbers” are not numbers of any kind. They are the result of dysfunctional minds.”
^ Ah yes, most of the classic mathematical proofs involve the author accusing their critics of having dysfunctional minds.
Please tell me you’re THE John Gabriel — the guy who tried to prove that the real numbers are countable: link.
Yes, and he’s been at this for at least 8 years. Mathematicians should definitely ignore whatever he has to say from the get go. I know it’s not open-minded, but economically speaking, we’ll never gain anything, and there is a huge cost of time utilized in debating this crank.
Look at this, for example, and Ben’s link above:
http://en.wikipedia.org/wiki/Wikipedia:Votes_for_deletion/John_Gabriel%27s_Nth_root_algorithm
Notice “Octonion” has “onion” in its name.
Therefore it must make people cry.
Decrying complex numbers because they have an “imaginary” part doesn’t show an appreciation that the definition of “number” has constantly been expanded.
“Imaginary” is just a word and does not literally mean that expansion of the sense of what a number could be is any shadier than the words like “negative” or “irrational” are meant to cast suspicion on those kind of numbers.
The first numbers were used to count discrete objects: “One cow, two cows, three cows…” Under that most basic notion of a number, even the concept of zero is scary and unfamiliar – “What do you mean a number for having NO cows at all?!?!” Once you have the operation of addition, the opposite process of subtraction leads to the dangerous possibility of subtracting more than you have. Surely the idea of less that zero is the result of a deranged mind. But, not only can one contemplate the notion of subtracting 7 from 2, but once you tolerate the fanciful notion of a negative number you find lots of uses like accounting for assets and debits. A similar story applies to the irrational numbers, whose proposed existence was once routinely met with death threats (the Pythagoreans). Even fractions had the ring of absurdity about them once.
By the way, objecting to these expansions of the notion of number were not necessarily “wrong” in the strictest sense. After all, if a “number” is meant to keep track of your cows, it is not all wrong to object to the notion of “half a cow” or the even weirder suggestion of a “negative” cow. Expanding the notion of number is a willful decision to expand the scope of what you’re willing to deal with mathematically. A math with no negative numbers is a legitimate, albeit limited, mathematics. However, so long as each expansion of the idea of number is backwardly compatible with what numbers were used for before, nothing is lost and something may be gained.
The mathematical statement ” x^2 = -1 ” can be pondered. Nothing about the use of ” i ” undermines any of the math that existed in the real-only world. All that remains is to see if this expansion offers any useful gain. The applications in the real world are legion, such as in circuit theory. Complex number theory also aids solving integrals of real functions.
the simplest way to show the square root of i is to use logarithms. it is often forgotten that the logarithm of a complex number is a valid operation because of the polar notation that complex numbers can take on.
all complex numbers (z) can be represented with a magnitude (r) and argument / angle from the positive real axis (a).
z = r . e^(i . a)
z = r . (cos(a) + i . sin(a))
you could rewrite the exponential form to be entirely as a power of the natural base, therefore the natural logarithm can be defined for any complex number z:
z = e^(ln(r) + i . a)
ln(z) = ln(r) + i . a
to find the square root of a number using logarithms, you end up with:
x^2 = y
sqrt(y) = x = e^(ln(y) / 2)
substitute in the value i for y resulting in:
sqrt(i) = e^(ln(i) / 2)
to find ln(i) we refer to the above rule; the complex number “0 + 1 . i” has an argument of Pi/2 and magnitude / radius of 1. therefore ln(i) = ln(1) + i . Pi/2, and our equation becomes:
sqrt(i) = e^(ln(i) / 2)
sqrt(i) = e^( (ln(1) + i . Pi/2) / 2 )
= e^( (0 + i . Pi / 2) / 2 )
= e^( i . Pi / 4 )
= (cos(Pi / 4) + i . sin(Pi / 4))
= sqrt(1/2) + i . sqrt(1/2)
to JG and anyone else thinking the same – before criticizing complex numbers and denouncing literally centuries of mathematical proof on the subject, I would like to hear your alternative to what the value sqrt(-1) means and its implications. if you have no issues with the existence of roots of negative numbers and can accept their mathematical treatment outside of complex numbers, your complaints about complex numbers have no basis – the expression i^2 = -1 can be made to be a substitution in the entire formulation of complex numbers, and it would not make a bit of difference to the results. what it would do is make the intermediate mathematics needlessly verbose.
but note that if you start denying the validity of roots of negative numbers, you will need to discard a lot of number theory along with it. much of what we know about the nature of numbers comes from studying the natural base and its counterpart, the natural logarithm – the trigonometric functions and the root of -1, along with the natural base e, share a relationship so deep (countless relationships and proofs exist) that if you are not prepared to accept it, i’m sorry to say, you’re the one with the dysfunctional mind here.
For me the beauty of “i” comes from the relationship: -1=exp(i*pi). All the three “dysfunctional mind” numbers are cross related: e, pi, & i. Without “i”, “e”and “pi” would not exist.
Moreover, z=x+iy=r*exp(i*theta); where r=sqr(x^2+y^2) & theta=atan(y/x)
From here, you can get
sqr(sqr(-1))=(-1)^{1/4}=
sqr(i)=exp(i*pi/4)=
cos(pi/4)+sin(pi/4)=
sqr(2)+sqr(2)i
🙂
John, I can assure you that complex numbers were invented by mathematicians, and that they form a major and fundamental part of mathematics.
But… what’s the square root of negative i??? I’m not very good in math, but i feel that can’t be a complexe number…
P.s: Very good blog… i am czech, so my english isn´t good…
The square root of “negative i”: well you can get that by multiplying the square root of i (shown above) by i itself. Why ? Because if we abbreviate the square root above to being “x”, the square root of negative i (multiplied by itself) would then be ix*ix = i*i *x*x and since x is the square root of i, x*x = i, and the i*i=-1 , multiplied together giving -i, so the original quantity, ix, must have been the square root.
Simply square 1+i gives (1+i)(1+i )=1+i+i-1 =2 i
so 1+i divided by root 2 must be the root of i.
For more generality, squaring x+ iy gives x^2 – y^2 +2ixy where x^2 -y^2 gives a real part A, and 2xy multiplies the i giving imaginery part iB. Its easy to choose x and y to arive at WHATEVER complex number A+iB after…… a bit of trigonometry 0r algebra
x^2-y^2=A
2xy=B
thus y= B/2x
so x^2-(B/2x)^2 =A
and multiply by x^2 gives
x^4 – Ax^2=B/4
A quadratic in x^2 and so
x^2 = A plus or minus sqrt(A^2 +B) gives our x ….and now y is easy.
Also its possibly a litttle better to use polar coordinates as in earlier post but here we show it can be done directly.
Who is this John Gabriel, and why does he expect to be taken seriously, and his ideas to be respected when he does little but go on about how brilliant he is, how stupid some of the greatest mathematicians in history were, and how retarded his critics are?
John … a little POLITENESS and HUMILITY go a long way.