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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
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.
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.
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.
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)....
Why do you use complex numbers?
Complex numbers can help visualize physical effects like the electromagnetic fields
around wires carrying current. Refer to the link below.
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.
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.
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)
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).
Is Tim meeker real or imaginary?
Tim Meeker is a character based on the real person Tim Meeker. If you read the Epilogue of "My brother Sam is Dead" It explains that historical records support much of the story, like most of the characters. The characters of the book are set to act as the author believed the real people would have in that type of situation. Therefore, to answer your question, Tim Meeker was a real person that died, but the story is fake. The execution of his brother Sam was actually how a soldier named John Smith died. A good deal of the story is based on historical records.
What careers use complex numbers?
Engineering (especially electrical engineering), Math, Physics mainly.
The sum of two complex numbers is always a complex number?
A "complex number" is a number of the form a+bi, where a and b are both real numbers and i is the principal square root of -1.
Since b can be equal to 0, you see that the real numbers are a subset of the complex numbers. Similarly, since a can be zero, the imaginary numbers are a subset of the complex numbers.
So let's take two complex numbers: a+bi and c+di (where a, b, c, and d are real). We add them together and we get:
(a+c) + (b+d)i
The sum of two real numbers is always real, so a+c is a real number and b+d is a real number, so the sum of two complex numbers is a complex number.
What you may really be wondering is whether the sum of two non-real complex numbers can ever be a real number. The answer is yes:
(3+2i) + (5-2i) = 8.
In fact, the complex numbers form an algebraic field. The sum, difference, product, and quotient of any two complex numbers (except division by 0) is a complex number (keeping in mind the special case that both real and imaginary numbers are a subset of the complex numbers).
Why were imaginary numbers created?
See related link for more information.
Originally, they were created so that every polynomial would have a solution. For example, the polynomial x² - 1 = 0 is a second order polynomial, so it should have 2 solutions. It does have 2 real solutions: 1 & -1. You can graph y = x² - 1 and see where the graph crosses the x-axis (these are the x coordinates that make y=0 and satisfy the equation). But what about x² + 1 = 0. If you graph y = x² + 1, it does not cross the x axis, but every polynomial is supposed to have a number solutions equal to the order {2nd order should have 2 solutions, 3rd order should have 3 solutions, etc.}
To handle polynomials like this, a number i was created such that i² = -1. Now this number i can be used to solve x² + 1 = 0. The solutions are x = i & -1. For many years, these numbers were considered just an imaginary concept, and for not much use until the work of Euler related them to sines and cosines. Now, imaginary and complex numbers are used to express the relationships between waves (in particular, electromagnetic waves and alternating current electricity).
In advanced calculations.
Imaginary numbers are used extensively in the development of electronic systems, control systems and physics.
The product of the complex numbers 4 3i 3 - 4i?
The easiest way to do this is treat i like a variable and multiply the binomials and combine like powers of i, then anywhere there is i², substitute it with (-1).
So (4 + 3x)(3 - 4x) = {using FOIL} 4*3 - 4*4*x + 3*3*x - 3*4*x² = 12 -7x - 12x²
So with x = i, you have 12 - 7i - 12*(i²) = 12 - 7i - -12 = 24 - 7i
One quick check to see if there is an error: The magnitude of the product of the two complex binomials will equal the product of the magnitude of each factor.
Magnitude of a + bi = sqrt(a² + b²). The magnitudes of two numbers that are multiplied are: sqrt(4² + 3²) = 5, and sqrt(3² + (-4)²) = 5. The magnitude of the answer is sqrt(24² + (-7)²) = sqrt(576 + 49) = sqrt(625) = 25, which is 5 times 5.
How do you work out the Arg of a complex number?
A complex number z = x+iy or (x, y) can be also represented as (r, θ) in polar coordinate, where r = √(x2+ y2) and θ = tan-1(y/x). Here θ is known asArg(z). And the values of θ in
]-π, π] is known as principal value of the argument and is represented as arg(z). It is evident that Arg(z) = arg(z) + 2nπ.
