0
1. i1 = i, i2= -1, i3= -i, i4= 1, i5= i, i6= -1 and so on.
2. Euler's constant e, when raised to the power of a complex number, gives a point in the complex plane according to Euler's Identity, e^(iθ
) = cos(θ
) + i*sin(θ
).
Whenθ
is replacedby pi, we get:
e^(iπ)
= cos(π
) + i*sin(π
)
= -1 + 0
= -1
In other words, e^(iπ)
+ 1 = 0.
3. Many intriguing fractals have been discovered in the complex plane, the most famous of which is the Mandelbrot Set. This is formed by an iterative process of a point in the plane. It produces a beautiful image, nicknamed by many as the "Thumbprint of God". (Google Images is your friend here - check it out.)
4. De Moivre's Theorem holds true in the realm of the imaginary, stating that
(r cisθ
)n= rncis nθ
.
This is a fascinating theorm in itself, but it also has hidden implications.
For example, if r = 1 (the point was 1 unit away from the origin), and you were to plot this point with a small angle above the positive real axis, then raising this number to an increasing power would result in that point making a full revolution about the origin and coming right back to where it started.
This is because rnstays the same (1^(anything) = 1), but the angle is slowly increasing as the power does. Therefore, as the angle increases, a circle is plotted around the origin with radius 1.
There are many other fascinations around complex numbers, but this covers the basics.
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What are some imaginary numbers?
i is the Imaginary Unit, equal to sqrt(-1). So i and any real number multiplied by i will all be imaginary numbers. Here are some: i, -i, 5i, -3i, i*pi, etc.
When were imaginary numbers invented?
Rafael Bombelli defined imaginary numbers in 1572, and Descartes named them 'imaginary' in 1637. It wasn't until the work of Euler in the 1700's that a usefulness for imaginary numbers was found, though. See the Wikipedia articles I linked for some good information on imaginary and complex numbers. I also linked an explanatory video that is pretty good as well.
How did imaginary numbers impact mathematics?
Imaginary numbers are used in complex numbers that make some math simpler like electronics where there is a cycle frequency it makes the math much simpler to handle complex equations
What is the difference between square roots and real square roots?
In mathematics, there is a distinction between real numbers and imaginary numbers. There is a number known as i which means, the square root of minus one. Since any number that we know of will produce a positive result when multiplied by itself, there would seem to be no such thing as the square root of minus one, however, the concept is useful for certain purposes, nonetheless, and it is therefore known as an imaginary number. Any multiple of i is also an imaginary number (such as 67i and so forth). So, some square roots are real numbers, and some are imaginary. Both types can be called square roots.
What are the differences between real numbers and imaginary numbers?
The set of real numbers is not closed under powers. That is to say, there are some equations of the form y = xa which do not have a solution within the set. Typical example: x is negative, a = 0.5
