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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.
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.
Every integer is a complex number?
Yes. Every integer is a rational number. Every rational number is a real number. Every real number is a complex number.
The complex numbers include all real numbers and all real numbers multiplied by the imaginary number i=sqrt(-1) and all the sums of these.
Imaginary numbers why do you need a positive and a negative?
Because an imaginary number is impossible otherwise. For instance, the square root of negative nine (-9) is an imaginery number because any two numbers multiplied by each other yield a positive number. So the SQR of -9 must have a rational part (SQR(9)) and an imaginary part, which assigns the negative.
A:Computers work on a binary system, and western maths is based on + and _. But if there was a third category, called neutral, then the square root of minus one would be neutral 1. And the whole strange notion of imaginary numbers would be unnecessary. There are questions which can't be answered by 'yes' or 'no', when neither is applicable. In China, the answer to a question such as "Have you stopped beating your wife?" would be wumu, meaning both yes and no or neither. On a 2-dimensional graph, + is to the right, - to the left of the upright line. And neutral sticks up off the paper from zero to your eye. In a third dimension.The concept of imaginary numbers doesn't exist in China, because they think differently. We can put weights on both pans of a balance (back weighing). Or, if you have a series of rooms, each with normally always two chairs, and then take one away in one room, we would say that room now has one chair. But in China, they would say it has minus one, since it is one less than normal. It is merely a different way of thinking.
Correction:The concept of imaginary numbers does in fact exist in China, and pretty much everywhere else that has bothered to investigate along these lines of inquiry. China, a major contributor to international math, especially over at least the last 100 years, is completely aware of the concept and application of imaginary numbers.Additionally, because something has two component parts does not make it an example of westernized dichotomism as a philosophy; it simply means something has two (or one, or three) parts -- nothing more.
As to computers having two states: zero and one, this is an artifact of the means by which technology evolved, specifically electrical states of off and on, which are easy to detect, as opposed to analog electrical states, which require a lot more control and instrumentation (consider the relative complexity of a voltage meter versus a wall switch as an example). I should add that China, who is now accountably a world leader in some aspects of computer systems design, is certainly adept and comfortable with binary math and boolean algebra.
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 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.
An imaginary number is a number that cannot exist. An example of an imaginary number would be: the square root of negative nine, or any negative number.
When I try to think of any two of the same numbers that would multiply together to be negative nine, all I can think of is 3 or -3. when I square both of those numbers, I get the number 9, not -9. When I multiply two negatives together, I get a positive number, therefore there is no possible way to get the square root of -9, or any negative number.
What is the POLAR form of complex numbers?
By creating a real-imaginary plane (real on horizontal axis, imaginary on vertical), any complex number can be represented graphically. The polar form is a magnitude and angle. The magnitude is measured from the origin to the point on the plane. For a complex number a + bi, this value is a2 + b2. The angle is measured from the positive real axis, clockwise.
For positive imaginary part (b), this will be +arccos(a/(a2 + b2)). (0° to +180°, or 0 to +pi radians)
For negative imaginary part (b), this will be -arccos(a/(a2 + b2)). (0° to -180°, or 0 to -pi radians, or alternatively 180° to 360° or pi to 2pi radians)
The answer depends on the context, but bi is a variable, not a number. It could be the ith of a set of variables b1, b2, ... .
Or it could be the square root of -b2.
Or it could be a vector of magnitude b in the idirection.
How do you find out the number of imaginary zeros in a polynomial?
Descartes' rule of signs (see related link) can help you determine the maximum number of real roots. If the polynomial is odd powered, then there will be at least one real root. Any even powered polynomial can be factored into a bunch of quadratics [though they may not be rational or even pretty], and any odd-powered polynomial can be factored into a bunch of quadratics and one linear (this one would have the real root). So the quadratics may have pairs of real or complex roots (having an imaginary component).
To clarify, when I say complex, I'm referring to the fact that there will be an imaginary component to the root, because actually the real numbers is a subset of the set of complex numbers.
The order of the polynomial will tell you how many roots it will have. If you can graph the polynomial, then you can see if it crosses the x axis. If it is a 5th order polynomial, and crosses the x axis 3 times, then there are 3 real roots (the other two roots are complex).
A complex number is a number with a real and an imaginary part.
Look at the links I will place below to find out more
A complex number is any number that can be represented as the sum of some number on the real number line and some number on the imaginary number line, a number line perpendicular to the real number line that contains multiples of sqrt(-1), which is more commonly denoted as i.
How do you rationalize complex numbers?
you must multiply by the conjagate. which is the denominator with the middle sign changed....(5+6i)...conjagate= (5-6i)....
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.
