Best Complex Numbers Quotes

"The discovery of the cubic formula led to the discovery of complex numbers by Rafael Bombelli not long after Cardano published the cubic formula."

"Imaginary numbers discovered as a quirky, intermediate step on the way to solving the cubic turn out to be fundamental to our description of reality."

"Complex numbers are just two real numbers added together with I."

"Every complex number can be represented as a point on the XY plane."

"Complex conjugating a product requires you to complex conjugate both pieces of the product."

"Imaginary numbers do exist, and they have their place in understanding the universe."

"Complex numbers are unreasonably effective in mathematics."

"Multiplying two numbers on the circle will give us another number on the circle."

"If we take the complex number at 45° and multiply it with the one at 90°, the result will be the one at 135°."

"These are called roots of unity: complex numbers that contain pure direction, like -1 or i."

"...as soon as we define a real function, we can actually already evaluate that function on the complex numbers as well."

"If a complex valued function merely has a first derivative on some open region of the complex plane, then it automatically has all higher order derivatives on that region too."

"Functions in the complex plane often behave much more nicely than functions on the real line as there are fewer edge cases to worry about when working with them."

"This integral over the boundary of d r, that's 2 pi i root 3 minus root 2."

"Euler’s formula is the gateway to study more about complex numbers."

"Adding two complex numbers simply means adding the real and imaginary parts separately."

"One of the best things about complex numbers is how they provide a really elegant way to describe and manipulate rotations in two Dimensions."

"Squares now mean multiplied by complex conjugate."

"We can recognize that this particle is just moving in a ring, in a circle. Of course, that should remind you of complex numbers."

"Now this imaginary number deal might not fit in the real number line, but they can be depicted of having an axis of their own."

"...how complex numbers can succinctly describe geometric relationships and provide a really natural and elegant language to describe 2D rotational Dynamics."

"Once you get used to them, complex numbers are actually one of the most elegant and ironically simple to use tools in your mathematical Arsenal."

"Complex numbers are the language of 2D rotation."

"Dividing by a complex number does the opposite, a rotation by the same angle but in the opposite direction."

"The complex numbers are really nice like that but that being said this equation has two very simple solutions to find."

"The reason why we choose e to the ix, it's easier to work with but then at the very end we can take the real part and this will produce a cosine."

"An example of a complex number would be 2 + 3i, and you can think of it as walking away from zero, first by 2 in the positive direction, and then by 3 more in the Imaginary direction."

"If a bi Vector squares to a negative number, its exponential performs a rotation."

"The move to complex numbers was the biggest jump in the history of math."

"Now in the 19th century this was well understood, Gauss proved the fundamental theory of algebra, they knew that this should be true, they worked in the complex numbers."

"You've just multiplied and divided two complex numbers."

"So if you think about complex plane, here's the real part of psi and here's the imaginary part of psi. Complex conjugation takes a point here and just moves it all the way down to a point right down there."

"... imaginary numbers and more general complex numbers... are bread and butter of modern mathematics and modern science."

"Our best theories of the physical reality are based on quantum mechanics and in quantum mechanics complex numbers are really indispensable."

Extending complex numbers: "People tried to find a three-dimensional version of the complex numbers, but Hamilton found a four-dimensional version called the quaternions."

"If we have two quantum state vectors, Psi and Phi, such that one of them is equal to the other one multiplied by some complex number alpha, then we say that the states differ by a global phase or alternatively that they're equivalent up to a global phase."

"Imaginary numbers are real numbers multiplied by the imaginary unit I, which is equal to the square root of minus one."

"The complex numbers are algebraically closed."

"We're about to enter the realm of complex numbers."

"In the complex plane, there are certain special places where when you put those complex numbers into the function, the function goes to zero."

"Everything in the complex plane can be written in polar form."

"We want to show that e to the i theta is cosine theta plus i sine theta."

"This is of the form \( e^{i\theta} \) where \( \theta \) is natural log of \( n \) times \( b \)."

"D'Morse theorem essentially allows us to take the power of complex numbers and the roots of complex numbers."

"If Z is equal to R cosine theta plus I sine theta is a complex number and N is any positive integer, then Z to the N is equal R to the N cosine N theta plus I sine N theta."

"Every real number has a complex number that's a square root."

"Multiplication of complex numbers on the circle is the same as adding their angles."

"Complex numbers are really good for rotations of the two-dimensional real plane."

"Euler's identity: e to the i y is equal to cosine of y plus i times sine of y."

"Euler's formula: e raised to the power I theta is the same thing as cosine theta plus I sine of theta."

"Complex numbers have a real component and an imaginary component."

"Complex or imaginary numbers arise naturally in solving polynomial equations."

"Euler's identity e to the ix is cos x plus i sine x is one of the most remarkable equations in the whole of mathematics."

"Imaginary numbers... they're not called imaginary because they don't actually exist."

"These numbers are an essential part of mathematics, with a whole branch of mathematical analysis being dedicated to them."

"Why did it take so long for mathematicians of the past to recognize the value of complex numbers?"

"The squared magnitude or the squared length of the complex number is the square root of a squared plus b squared."

"There is no real solution to the square root of negative 4."

"If you enlarge numbers to include complex numbers, then you can solve any polynomial equation."

"There are very few branches of any science where complex numbers will not be used."

"If you have this negative I, you're going to have the complex conjugate also as a zero."

"I'm motivated for you why we care about imaginary numbers and how useful they are in real math, even beyond algebra, in real engineering and science."

"You've got a complex number here and a complex number here; you multiply these two complex numbers, whatever path you go through in the woods, you should end up with another complex number."

"We multiply the moduli, we add the arguments."

"By definition, a complex number is simply any number in the form a plus ib, where a and b are real numbers."

"Z equals R e^(i theta), where R is the modulus and theta is the argument of Z."

"Not all complex numbers are the real numbers."

"So a number in that form is a complex number."

"The field of complex numbers has been constructed from R2 using vector addition."

"So you can visualize complex numbers as vectors in R2."

"A plus iB can be taken to be the natural basis elements."

"Complex solutions always come in conjugate pairs."

"E to the i-x is cos x plus i sine x, which is my identity, one of the most remarkable equations in the whole of mathematics."

"The absolute value of a plus bi is equal to the square root of a squared plus b squared."

"The final answer is negative 6 radical 2i."

"The final answer is negative 10 minus 198 times i."

"The final answer is negative 9 plus 40i."

"The final answer is 48 divided by 37 minus 8 over 37 times i."

"So this is equal to negative two minus five i."

"Whatever expression you have that's a complex number, you can always put it in standard form."

"This is like saying I have \( W_n^r \) and \( W_n^{r + \frac{n}{2}} \), and this guy is just equal to negative one."

"The square root of -1, i, is the most important single invention in all of mathematics."

"When you multiply complex numbers geometrically, you just multiply the lengths and add the angles."

"It's a good idea to think about them in these polar coordinates, r e to the I Theta."

"Complex numbers are extremely important in differential equations."

"The bin values are complex numbers because they represent both magnitude and phase."

"Every complex number z can be written as its absolute value times some phase."

"There's a very famous identity due to Euler which says that e to the ix is equal to cosine x plus i sine of x."

"A complex function is a function where we're allowing complex numbers to be plugged into the function and complex outputs to come out of the function."

"Complex roots occur in conjugate pairs."

"If one of my roots is 2 plus 3i, my other root is simply 2 minus 3i."

"Complex numbers are a combination of real numbers and imaginary numbers."

"We have one more lesson where we will wrap it up with dividing complex numbers."

"Taylor series are also really nice because they generalize to complex valued functions really, really well."

"The power of complex numbers is they turn differential equations into algebraic equations."

"The complex conjugate is the same thing as the denominator with the opposite sign."

"Multiplication of complex numbers corresponds to the addition of the two angles and the multiplication of the lengths."

"When I have complex eigenvalues, I'm probably going to be getting things like cosines and sines as my solutions."

"E to an imaginary number X will give me cosine X plus I sine of X."

"We can view the set of complex numbers as a plane."

"Complex numbers have a real part and an imaginary part, giving us a nice graphical interpretation."

"The correct product is actually 10 plus 10i."

"If K is not equal to L, then this sum is equal to 1 minus e to the 2 pi K minus L divided by n raised to the nth power."

"Multiplying Phi by Phi star to get Phi squared will only add e to the I alpha times e to the minus I alpha which is equal to 1."

"When you multiply a number by its complex conjugate, you actually end up with just a real number."

"The multiplication of complex numbers just multiplies their magnitudes and then you linearly add their angles together."

"The easiest way to look at powers of complex numbers is probably to write out all these Z's using Euler's formula."

"The distance D is actually the modulus of the complex number Z minus z0."

"Complex number you can think of as a point in the plane."

"Complex numbers, which have an amplitude and phase, are therefore perfect for this."

"Every complex number of absolute value one is just beautifully written in the form e to the i theta."

"The complex exponential function \( e^{z} \) is defined by \( e^{x} \times e^{iy} \), where \( z \) is \( x + iy \)."

"The standard form of a complex number is a plus bi, where the first value must always be your real number, and then the imaginary part will always be last."

"The set of complex numbers is all numbers of the form a plus bi, where a and b are both real numbers, and i squared is negative 1."

"The argument of the complex number is the angle that is made by the ray defined by this complex number and the positive real axis."

"Euler's formula is e to the i theta is equal to cosine theta plus i sine theta."

"The fundamental theorem of algebra guarantees for us that there's this gigantic bed, namely the field of complex numbers, over which any polynomial will split into linear factors."

"Recall Gaussian integers are a subset of the complex plane; they form a Euclidean domain."

"If P equals a squared plus b squared, then a plus or minus bi are primes in the Gaussian integers."

"We have primes of the form a plus or minus bi when n is equal to P, which can be written as a squared plus b squared."

"A complex number is just a number that has a real part here and an imaginary part here."

"In my playground, it's all about e to the i j Omega t."

"Probability amplitudes are complex numbers which means that they can interfere and cancel one another in ways that normal probabilities can't."

"There's nothing imaginary about complex numbers at all; they very much exist in the real world and have applications in thermodynamics, mechanical engineering, and electrical engineering."

"If negative numbers and zero were strange concepts for humanity to wrap its collective head around, then imaginary numbers were downright mind-boggling."

"Every answer can be put in a plus bi form, meaning you've got a real component and an imaginary component."

"The beauty of complex numbers is that i^2 is -1, always."

"J has a definition: J is equal to the square root of minus one."

"I squared is negative 1. Always, always, always."

"Under what conditions will the product of two complex numbers always be purely imaginary? The real component must be equal to zero."