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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 do you calculate the number of parking spaces need for an office complex?
You need first to know the ratio ofvehicles percapita of your city. Then you'll need to know what kind of people will work in this complex. You know, one hundred managers or lawyers will surely have (and use) more cars than one hundred call center operators. The third data to get is the availability of public transportation around this complex, if it is too difficult to get by train and/or bus, people will surely use more cars (or bicycles depends in witch country you're in). That's all for the employees of the complex. The next step is to calculate the flow of people visiting this complex, to know what kind of people and so on.
Usually in bigger cities the parking is paid (and expensive) and office buildings/complexes have far more parking spaces than the building (or buildings) alone would need, just to make a profit with the lack of parking spaces in the neighborhood.
Who invented complex numbers and when did he invented?
The Italian mathematician, Gerolamo Cardano conceived of complex numbers around 1545.
Is every natural number is a real number?
Yes, natural numbers are the set of "counting numbers" - integers bigger than zero. Hence they are all real numbers.
A triangle, with one of the complex numbers represented by a line from the origin to the number, and then move from that point up and over the amount of the next complex number. Then draw a line segment from the origin to the final point.
Which number does not belong in the series 2-3-6-7-8-14-15-30?
8 does not belong
Sequence is to add 1 for the first number, and double the next result, and then add 1 to that result, and double again...
Correct Sequence
2 (+1) 3 (x2) 6 (+1) 7 (x2) 14 (+1) 15 (x2) 30
2-3-6-7-8-14-15-30
(2+1) = (3 x 2) = (6 + 1) = (7 x 2) = (14 + 1) = (15 x 2) = 30 ....
So 8 does not belong in the above series.
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.
An imaginary number is a number that has the square root of -1 as one of its factors.
How are imaginary numbers used in electricity?
The very simplified answer is that imaginary numbers put together with real numbers (to make a complex number) can describe the timing of voltage relative to current, or current relative to voltage, in an AC circuit. Let's say that we're driving an AC electrical circuit with an oscillating current source, and measuring a resulting oscillating voltage. Here's the rub:
Purely Real: If you put a resistor in the circuit and measure the voltage oscillations across it, the voltage will be a purely real number. This means that the timing of the voltage peaks will match the timing of the current peaks exactly.
Purely Positive Imaginary: Now, put an inductor in the circuit instead of a resistor and measure the voltage oscillations. It will be a purely positive imaginary voltage. This does not mean that the voltage is non-existent (as many people think)! It simply means that the voltage peaks will be one quarter cycle ahead of the current peaks, or 90 degrees ahead. The voltage has physical value. If you were to touch the ends of the inductor, you would still get shocked! The imaginary property just tells you that the timing is ahead by a quarter cycle, that's all--nothing esoteric or "complicated." A good analogy to this would be if you were riding your bicycle side by side with your friend, and you were pedaling at the same rate, BUT your pedal was consistently a quarter turn ahead of his.. Your timing could be considered purely imaginary relative to him (or her).
Purely Negative Imaginary: Now, put a capacitor in the circuit and measure the voltage oscillations. It will be a purely negative imaginary voltage, which simply means that the voltage peaks will be one quarter cycle behind of the current peaks, or 90 degrees lagging.
Complex: By putting a combination of resistors, inductors, and capacitors in the circuit together, you get a complex voltage, allowing you to get "in between" values. For example, you could carefully size a resistor and inductor, put them in series, and force the voltage peaks to be 45 degrees ahead.
Hope this is clear. If it's still cloudy, I'll paste a link in the web link area that has a site out there with an interactive explanation showing how imaginary numbers can be used with complex numbers to represent both size and timing (it's actually my site, but for educational purposes only).
While these answers mainly deal with electric power [alternating current], the same concepts apply to waves in general which have a phase difference [difference in timing of peaks and valleys of the waves].
Please see the below link for a graph of the fields around current carrying
conductors by the formula: w=(z-1)/(z+1), z=x + iy.
How do you convert the complex number minus i into polar form?
A COMPLEX NUMBER CAN BE CONVERTED INTO A POLAR FORM LET US TAKE COMPLEX NUMBER BE Z=a+ib a is the real number and b is the imaginary number THEN MOD OF Z IS SQUARE ROOT OF a2+b2 MOD OF Z CAN ALSO BE REPRESENTED BY r . THEN THE MOD AMPLITUDE FORM IS r(cos@
Very interesting, but -i is not a complex no. it is a simple (imaginary) no. with no real part.
What does Euler's formula equal?
Euler's formula states that, for any real number x,
eix = cos x + i sin x,
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine, respectively. The argument x is given in radians.
Please see the related link below for more information.
How do change a complex number to its standard form?
It isn't clear in what form you have the complex number. But you can change it from the form (absolute value, angle) to the form (real part + imaginary part) using the polar-rectangular conversion available on scientific calculators (and the other way round, with the rectangular-polar conversion). Note that a complex number in the form (real part + imaginary part) is most appropriate for addition and subtraction, while a complex number of the form (absolute value, angle) is most appropriate for multiplication or division, so depending on the operations, you may want to convert back and forth several times.
What is complex number system?
All pairs of numbers, written in the form a + bi (for example: 3 + 5i, or 7 - 2i, etc.), where the first number is called (for historical reasons) the "real part" and the second number the "imaginary part". Complex numbers can be graphed as points on a plane. They have important applications in several fields of science, arts, and pure mathematics.
What is the name for the number 1000000000000000000000?
Yes -- one sextillion.
Also according to one scientific hypothesis, this is the estimated number for the number of stars within our current universe with the number slowly increasing as time progresses.
Is the sum of two pure imaginary numbers always a pure imaginary number?
Yes, the only argument would be the example, i + (-i) = 0. However, many people don't realize that 0 is both a purely real and pure imaginary number since it lies on both axes of the complex plane.
What is the exponential form of complex numbers?
Complex numbers can be represented as reiƟ, where r is the distance from the origin, and Ɵ is the angle (in radians) with the positive real-axis (horizontal axis).
What are the roots of complex numbers in mathematics?
See the answer to the related question: 'How do you solve the power of an imaginary number?' (Link below)
How do you determine the absolute value of a complex conjugate?
the absolute value of x + iy is equal to (x^2+y^2)^.5
and is the same for the conjugate, x-iy
What is the largest number of pieces you could cut a pie into with 4 straight cuts of a knife?
This depends on whether the pie is thick or not - if you mean 4 cuts from the top downwards, then you can get nine, if you cut it right. Imagine a clock face. cut 1 - from 1:00 to 5:00 cut 2 - from 11:00 to 7:00 cut 3 - from 10:00 to 2:00 cut 4 - from 8:00 to 4:00 Alternatively, with a thick pie you could make three top-downwards cuts to give six people and one through the middle of the pie, parallel to the plate, to divide each of those pieces in two - giving a total of 12 pieces. (This works when if you're dealing with 'cake' instead of 'pie'.)
Find the square root of a complex number system?
Probably the best way is to change the complex number to its polar form, [or the A*eiΘ form] A is the magnitude or the distance from the origin to the point iin the complex plane, and Θ is the angle (in radians) measured counterclockwise from the positive real axis to the point. To find the square root of a number in this form, take the positive square root of the magnitude, then divide the angle by 2. Since there will always be 2 square roots for every number, to find the second root, add 2pi radians to the original angle, then divide by 2.
Take an easy example of square root of 4. Which we know is 2 and -2. OK so the magnitude is 4 and the angle is 0 radians. zero divided by 2 is zero, and the positive square root of 4 is 2. Now for the other square root. Add 2pi radians to 0, which is 2pi, then divide by 2, which is pi. pi radians [same as 180°] points in the negative real direction (on the horizontal), so we have ei*pi = -1 and then multiply by sqrt(4) = -2.
Try square root of i. i points straight up (pi/2 radians) with magnitude of 1. So the magnitude of the square root is still 1, but it points at pi/4 radians (45°). Converting back to rectangular gives you sqrt(2)/2 + i*sqrt(2)/2. The other square root will always point in the opposite direction [180° or pi radians]. So the other square root is at 225° or 5pi/4 radians, and the rectangular for this is -sqrt(2)/2 - i*sqrt(2)/2. Using FOIL (from Algebra) you can multiply it out like two binomials and you will get i when you square either of the two answers for square root.
What is a multiplicative inverse of an imaginary number?
The same as for a real number: 1 divided by the number.For example, the multiplicative inverse (or reciprocal) of 2i is 1 / 2i = -(1/2)i.
How do you solve i25 when i is an imaginary number?
i25 · Step 1. 25÷ 4 has a remainder of 1 · Step 2. i25 = i1=i Rule : i4 is equal to 1 apply the following formula to reduce i to any power just divide the given power by 4 and then whatever you get as the remainder just put it as the power of i in your answer ik = ir Where r = the remainder of k÷4 And then solve it like i1 would be i or i2 would be -1 or i3 would be -i and so on. Other Example: i9 § Step 1. 9 ÷ 4 has a remainder of 1 § Step 2. i9 = i1 i103 § Step 1. 103 ÷ 4 has a remainder of 3 § Step 2. i103 = i3 = -i
Why and what is principle argument of complex number?
A complex number (z = x + iy) can be plotted the x-y plane if we consider the complex number the point (x,y) (where x is the real part, and y is the imaginary part). So once you plot the complex number on the x-y plane, draw a line from the point to the origin. The Principle Argument of z (denoted by Arg z) is the measure of the angle from the x-axis to the line (made from connecting the point to (0,0)) in the interval (-pi, pi]. The difference between the arg z and Arg z is that arg z is an countably infinite set. And the Arg z is an element of arg z. Why? : The principle argument is needed to change a complex number in to polar representation. Polar representation makes multiplication of complex numbers very easy. z^2 is pretty simple: just multiply out (x+iy)(x+iy). But what about z^100? This is were polar represenation helps us, and to get into this representation we need the principle argument. I hope that helped.
Who are the mathematicians that discovered imaginary numbers?
The first person to write about them was Gerolamo Cardano in 1545, but he doesn't seem to have taken them seriously.
See http://en.wikipedia.org/wiki/Gerolamo_Cardano .
The first serious use of imaginary numbers (better, complex numbers) was by Rafael Bombelli, published in 1572. He used them as intermediate steps when solving cubic equations.
See related link.
Is every real number a complex number?
A complex number is a number of the form a + bi, where a and b are real numbers and i is the principal square root of -1. In the special case where b=0, a+0i=a. Hence every real number is also a complex number. And in the special case where a=0, we call those numbers pure imaginary numbers.
Note that 0=0+0i, therefore 0 is both a real number and a pure imaginary number.
Do not confuse the complex numbers with the pure imaginary numbers.
Every real number is a complex number and every pure imaginary number is a complex number also.
