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Complex Numbers
The square root of negative one, which we now call the imaginary unit, was once thought to be an absurd notion. However, through the diligent studies of open-minded mathematicians, it was shown that the real numbers were actually just one part of a larger set of numbers known as the complex numbers, the other part being the imaginary numbers. Please direct all questions about these surprisingly useful and applicable numbers into this category.
887 Questions
How the set of real no is uncountable?
There is no one to one correspondence between the real numbers and the set of integers. In fact, the cardinality of the real numbers is the same as the cardinality of the power set of the set of integers, that is, the set of all subsets of the set of integers.
Not exactly. The numbers (a & b) can be any real number (positive or negative). It is the letter i, which represents the imaginary unit sqrt(-1).
False
How do you create a complex formula that calculates a 30 percent decrease?
We're stumped. The only formula up with which we're able to come is a simple one.
If the original quantity is 'Q', then after a 30-percent decrease, the new quantity 'N' is:
N = 0.7 Q
What is the relationship between a complex number and its conjugate?
When a complex number is multiplied by its conjugate, the product is a real number and the imaginary number disappears.
Which number belongs to the set of complex numbers?
Lots of numbers do.
To begin, all real numbers do.
Multiples of sqrt(-1), aka. imaginary numbers, do.
The Complex Numbers are all numbers which are the sum of a real number and an imaginary number.
How do you find cube roots of imaginary numbers?
For a pure imaginary number: {i = sqrt(-1)} times a real coefficient {r}, you have i*r. The cube_root(i*r) = cube_root(i)*cube_root(r), so find the cube root of r in the normal way, then we just need to find the cube root of i. For any cubic function (which has a polynomial, in which the highest term is x3) will always have 3 roots. There are 3 values, which when cubed will equal the imaginary number i:
- -i will do it: (-i)3 = (-i)2 * (-i) = -1 * (-i) = i
- The other two are complex: sqrt(3)/2 + i/2 and -sqrt(3)/2 + i/2
If you cube either of the two complex binomials by multiplying out, you will end up with 0 + i as the answer in both cases.
Note: the possible roots for any cubic are: 3 real roots, or 1 real root and 2 complex root, or 1 pure imaginary root, and 2 complex roots.
For your original question, if you want to stay in the pure imaginary domain, then you can use: Cube_root(i*r) = -i * cube_root(r) to find an answer.
Do all real numbers lie on a single line in the complex plane?
Yes. Traditionally, this line is drawn horizontally, with positive numbers to the right, and negative numbers to the left.
The set of complex numbers is the set of numbers which can be described by a + bi, where a and b are real numbers, and i is the imaginary unit sqrt(-1). Since a and b can be any real number (including zero), the set of real numbers is a subset of the set of complex numbers. Also the set of pure imaginary numbers is a subset of complex number set.
Is the product of two conjugate complex number always a real number?
Yes. This can be verified by using a "generic" complex number, and multiplying it by its conjugate:
(a + bi)(a - bi) = a2 -abi + abi + b2i2 = a2 - b2
Alternative proof:
I'm going to use the * notation for complex conjugate. Any complex number w is real if and only if w=w*. Let z be a complex number. Let w = zz*. We want to prove that w*=w. This is what we get:
w* = (zz*)*
= z*z** (for any u and v, (uv)* = u* v*)
= z*z
= w
Can x to the fourth power ever result in a negative number?
no even exponent of a real number can ever result in a negative number. If x is a complex number with the real and imaginary part having the same magnitude, then taking that to the fourth power will result in a real number, which is negative.
Example: (2 + 2i)4, or (-2 + 2i)4, or (2 - 2i)4, or (-2 - 2i)4, Just take (2 - 2i)4, as one to see how it works. First take (2 - 2i)2, then we'll square that result.
(2 - 2i)2 = 4 - 4i - 4i + 4i2 , but i2 is -1, so we have -8i, then square that is 64i2 which is -64.
What are the applications of complex numbers in real life?
- As with many other topics of advanced math, it depends whether your "real life" includes working in some engineering area, or not. If you work in electrical engineering or electronics, you will use complex numbers on a daily basis; similarly if you do a lot of math for some reason.
- Complex Numbers (ones involving the imaginary unit 'i' which is defined to handle the square root of -1) are a convenient way to describe the behavior of some electric circuits and waves.
- One thing you may encounter in your 'real life' are uninterruptible power supplies (UPS - what keeps your computer running when the power goes out). You will see ratings such as 500 VA / 300 W. This has to do with Complex Power. Electric circuits, which have inductances (such as transformers and electric motors) or capacitors, behave in a way that is described by complex numbers. While Volts X Amps equals Watts, the 300 W is the 'real power' (the amount available to do work), and the 500 VA is the apparent power (the total power that the power supply is capable of producing). The reactive power (due to capacitance & inductance) is represented by the imaginary component and it is at right angles to the real power (represented by the real component).
- Another use of complex numbers is in the Fast Fourier Transform (FFT) - one of the most ubiquitous algorithms - used heavily for signal processing. If you have a digital camera, a cell phone, an LCD - FFT is there, bringing complex numbers along.
- The complex equation w=z+1/z is a basic formula used for designing air foils-airplane wings and Figuring out flow forces around a circular object in water for instance.
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Are imaginary numbers rational or irrational?
Imaginary numbers are not intrinsically rational or irrational.
Of course, all real numbers are either rational or irrational numbers.
Imaginary numbers are not real numbers.
Imaginary numbers have a real part and an imaginary part, sometimes written like z=x+i y.
The two parts, i.e. the x and the y, are real numbers. As real numbers, they are either rational or irrational. Its just that the two parts of a complex number may both be either rational or irrational or one may be rational and the other irrational. One could always make up a new name for these cases, but right now there is no such classification.
Why must you rationalize complex numbers and not divide them?
Since a Complex number has a real component and an imaginary component, it would be like trying to divide z / (2x + 3y)
Can a no real number be a pure imaginary number?
If a number is pure imaginary then it has no real component. If it is a real number, then there is no imaginary component. If it has both real and imaginary components, then it is a complex number.
How do you enter an imaginary number in a scientific calculator?
Some scientific calculators can't handle complex or imaginary numbers. If you happen to have a special calculator that does, probably the manual will tell you how to enter them.
The HP 48 and up series does. It depends on if your calculator is in Polar Coordinate mode or X-Y coordinate mode, but a quick way to get the imaginary number i (regardless of which mode the calculator is currently in), is to press -1, then 'square root' button.
What is the Multiplicative inverse formula of complex numbers?
So if you have a number z = a + bi. Then how to find 1 divided by z. The way to figure this is to get the denominator as a pure real number. Multiplying the numerator and the denominator by the complex conjugate {a - bi} will result in a pure real denominator.
(a - bi)(a + bi) = a² + abi - abi - (bi)² = a² + b². So the multiplicative inverse is
(a - bi)/(a² + b²)
How do you use imaginary number in a scientific calculator?
Each calculator has its own nomenclature for working with imaginary and complex numbers. Many scientific calculators allow you to just type -1 and hit the square root button and it will give you something like (0,1) or (1,∠90°). In the first example, the first number {the 0} represents the real part, and the second number {the 1} represents the imaginary part. This is what happens on the HP-48 and HP-50 in Rectangular mode. In the second example, the calculator is in Polar mode (degrees), rather than Rectangular. So the first number {1} is the magnitude, and the second {90°} is the angle, measured in a counterclockwise direction from the positive real axis. 90° points straight up and is purely imaginary.
If my calculator was in radians mode, rather than degrees, then it would show (1,∠1.57) 1.57 radians is pi/2 (to 2 decimal places), which is the same angle as 90°.
An earlier calculator that I had, you first had to put the calculator in complex mode, then you had to push an extra button to view the imaginary part of the answer.
Find the complex conjugate of 14 plus 12i?
To find the complex conjugate of a number, change the sign in front of the imaginary part. Thus, the complex conjugate of 14 + 12i is simply 14 - 12i.
What is an example of rote counting?
Teacher"Show me how you counted to ten" Student "Like this:two,four,six"
