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
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
Are imaginary numbers real numbers?
When people started classifying numbers in different ways Some numbers were grouped together and called Real numbers. Solutions that would create Imaginary numbers were simply explained away as impossible, later the rules for working with these numbers, but, even though they are not considered Real numbers some math operations will create Real number answers.
Jobs that require the use of imaginary numbers?
Physics (e.g., quantum mechanics, relativity, other subfields) makes use of imaginary numbers. "Complex analysis" (i.e., calculus that includes imaginary numbers) can also be used to evaluate difficult integrals and to perform other mathematical tricks. Engineering, especially Electrical Engineering makes use of complex and imaginary numbers to simplify analysis of some circuits and waveforms.
What are complex or imaginary numbers?
An imaginary number is a number, which when squared, gives a negative real number. Any positive or negative real number, when squared, will give a positive real number. Imaginary numbers were originally conceived (around the 1500's) to provide solutions to equations which required there be a solution to the square root of a negative real number. Originally, that was the only purpose that they served, so they were given the term imaginary. The imaginary numbers were shown to be graphically at a 90° angle to real numbers. Complex numbers are the combination of real and imaginary numbers, and can be plotted graphically on a complex plane, just like you would plot x and y coordinates on a regular 2-dimensional plane.Through the work of Euler in the 1700's and others, a relationship between imaginary numbers and the behavior of waves and oscillating motion was worked out. See related link for some interesting information about imaginary and complex numbers.
What Numbers that can not be written as a ratio of 2 integers?
Some examples are an irrational number, an imaginary number, a complex number.
Are borders real or imaginary lines?
Some are real and some are imaginary
Is hydrogen in its own group?
Yes, hydrogen is in its own group on the periodic table. It is located at the top of Group 1, but it has unique properties that make it distinct from the other elements in that group.
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
Why is negativeroot not define?
The negative root is not defined because if you multiply two negative numbers together, the product is still positive. 2*2 =4 (-2)*(-2) =4 So, there is no real answer to the square root of a negative number. In higher math however, the square root of negative one is actually defined as i. The set of i is a set of imaginary numbers, and they have some special properties.
