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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
Evaluate and express this complex number in standard form the absulote value of 5-12i?
The absolute value of a complex number is it's magnitude (distance from the origin). Think about complex numbers graphically, with reals on the horizontal axis, and imaginaries on the vertical axis. Now you have a right triangle: From the origin move to the right 5 units, then move down 12 units. The absolute value, or magnitude, is the length of the hypotenuse.
For this triangle, it is 13: sqrt(5^2 + 12^2) = sqrt(25+144) = sqrt(169) = 13. For magnitudes, we are only interested in the positive square root.
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.
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.
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.
Are there more real numbers than imaginary numbers?
Both imaginary and real numbers are infinite .
Answer:Any real number can be turned into an imaginary number by multiplying it by "i" ot "j" (the root of -1). Hence it would appear that the set of all real numbers would equal the set of all imaginary numbers. However 0 (zero) multiplied by anything still equals zero. This would mean that there is at least one number that cannot be converted to an imaginary number.Is infinity an imaginary number?
no... because there is no such ting as infinity being a number ______________________________________________________________ Contributer 2: That's a right answer from contributer 1. But if you are really paranoid and desperately want to find out what infinity is, prepare to have your mind blown. This is not real #'s we are talking about. x/0 would equal infinity and -x/o equals negative infinity. We are assuming 0 is the threshold. This is not incorrect, simply very advanced.
How do you multiply imaginary numbers?
First, let's make sure we are not confusing imaginary numbers with complex numbers.
Imaginary (sometimes called "pure imaginary" for clarity) numbers are numbers of the form ai, where a is a real number and i is the principal square root of -1.
To multiply two imaginary numbers ai and bi, start by pretending that i is a variable (like x).
So ai x bi = abi2. But since i is the square root of -1, i2=-1. So abi2=-ab.
For example, 6i x 7i =-42.
5i x 2i =-10.
(-5i) x 2i =-(-10)= 10.
Complex numbers are numbers of the form a+bi, where a and b are real numbers. a is the real part, bi is the imaginary part.
To multiply two complex numbers, again, just treat i as if it were a variable and then in the final answer, substitute -1 wherever you see i2.
Hence (a+bi)(c+di) = ac + adi + bci + dbi2 which simplifies to ac-db + (ad+bc)i.
For example:
(2+3i)(4+5i) = 8 + 10i +12i + 15i2= 8 + 10i + 12i - 15 = -7 + 22i
The difference of two complex numbers is always a complex number?
A complex number is any number that is in the real/imaginary plane; this includes pure reals and pure imaginaries. The difference between two numbers inside this plane is never outside this plane; therefore, yes, the difference between two complex numbers is always a complex number. However, the difference between two numbers that are neither purely imaginary nor purely real is not always necessarily a number that is neither purely imaginary nor purely real. Take x+yi and z+yi for instance, where x, y, and z are all real: (x+yi)-(z+yi)=x+yi-z-yi=x-z. Since x and z are both real numbers, x-z is a real number.
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 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).
What is a set of complex number?
The related link shows a set of complex numbers that depict the Electro-Magnetic
fields around two wires. The formula is (z-1)/(z+1)
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.
Why do you need imaginary numbers?
because somtimes there isn't an answer to every equation like what's the square root of -16.... there is no answer so we would just use an imaginary number which is i.
It turns out that these are important in a practical sense. Imaginary numbers turn up all the time in quantum mechanics and certain types of electronic circuits as well.
How do you use imaginary numbers in real life?
Although most of us do not use imaginary numbers in our daily life, in engineering and physics they are in fact used to represent physical quantities, just as we would use a real number to represent something physical like the length of a stick or the distance from your house to your school.
In general, an imaginary number is used in combination with a real number to form something called a complex number, a+bi where a is the real part (real number), and bi is the imaginary part (real number times the imaginary unit i). This number is useful for representing two dimensional variables where both dimensions are physically significant. Think of it as the difference between a variable for the length of a stick (one dimension only), and a variable for the size of a photograph (2 dimensions, one for length, one for width). For the photograph, we could use a complex number to describe it where the real part would quantify one dimension, and the imaginary part would quantify the other.
In electrical engineering, for example, alternating current is often represented by a complex number. This current requires two dimensions to represent it because both the intensity and the timing of the current is important. If instead it were a DC current, where the current was totally constant with no timing component, only one dimension is required and we don't need a complex number so a real number is sufficient. The two key points to remember are that the imaginary part of the complex number represents something physical, unlike it's name implies, and that the imaginary number is used in complex numbers to represent a second dimension.
Remember, a purely imaginary voltage in an AC circuit will shock you as badly as a real voltage - that's proof enough of it's physical existence. I'll put a link in the link area to a great interactive site (it's actually my site but for it's educational purposes only) that explains the imaginary number utility more visually with animations.
Who invented imaginary numbers?
There is no one person who invented it there are several people who had contributed.
Can a complex number be a pure imaginary number?
Yes. And since Real numbers are a subset of complex numbers, a complex number can also be a pure real.
Another AnswerYes, for example: (0 + j5) is a complex number, whose 'real' number is zero.
Complex numbers, Real numbers, Rational numbers, Integers, Natural Numbers, Multiples of an integer.
