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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 find LCD in complex numbers?
I don't know if this will result in a least common denominator or not, but here is a system that you can use. Suppose you have the complex fraction: (a + bi)/(c + di) where {a,b,c & d} are all real numbers, and i is the imaginary unit number. What I would do is get the denominator to a real number, then use this same procedure for other fractions that you want to add or subtract, then find the LCD between those fractions will real-number denominators.
So the first step is to get the fraction to have a real-number denominator. Do this by multiplying the numerator and denominator by the complex conjugate of the denominator. The conjugate of (c + di) is (c - di), so (c + di)*(c - di) = c² - cdi + cdi + d² = c² + d², and (a + bi)*(c + di) = ac + adi + bci - bd = (ac - bd) + (ad + bc)i,so the new equivalent fraction equals:
(ac - bd) + (ad + bc)i
----------------------
c² + d²
Now do the same for the other fractions. You will have fractions in which all denominators are real numbers, then you can find LCD between these new equivalent fractions.
How do we apply complex number in our life?
If you've ever flown then you used complex numbers. The basic equation w=z+1/z is used to design air foils (airplane wings). While you don't actually concern yourself with these equations anymore than the thermodynamic equations that govern the running of your car's engine.
Check out the related link for some interesting application of imaginary and complex numbers, though.
complex to-complex(mag, theta)
{
if(mag >=0)
{
real = mag * cos(theta);
img = mag * sin(theta);
return real + i * img;
}
else
raise error;
}
One test is needed : mag must be a positive number!
And the return value is depending of your way to deal with complex number
If the discriminant is zero then there are no imaginary solutions?
Yes, if the discriminant is zero, then there will be a double root, which will be real.
Also, If the discriminant is positive, there will be two distinct real solutions. But if the discriminant is negative, then you will have two complex solutions.
Plot -5 plus 9i in a complex plane?
Start where the x and y axes cross.
Go 5 units to the left on the x (horizontal axis) and then go 9 units up from there.
Put a dot in that spot.
How do you find the additive inverse of a complex number?
You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).
You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).
You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).
You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).You take the additive invers of the real and of the imaginary part. For instance, the additive inverse of: (3 - 5i) is (-3 + 5i).
What did the king of the real valued functions say to the imaginary numbers?
The king said to get in step because they are 90 degrees out of phase.
What is the square root of negative 54 using imaginary numbers?
±3i√6
Rounded to two decimal places, the square root of +54 is equal to ±7.35. Therefore, the square root of -54, rounded to two decimal places, is equal to ±7.35 i.
Are skew symmetric roots purely real or purely imaginary?
They can be either. If they are roots of a real polynomial then purely imaginary would be symmetric and only real roots can be skew symmetric.
What is the magnitude of the complex number 3-9i?
Mag(3 - 9i) = sqrt(32 + 92) = sqrt(9 + 81) = sqrt(90) or 3*sqrt(10)
What is the definition of a pure imaginary number?
A pure imaginary number is a complex number that has 0 for its real part, such as 0+7i.
The square root of a negative value is called an imaginary or number?
Yes. The letter i denotes the value of the "positive square root" of -1. So i² = -1. But also (-i)² = -1 as well. Remember that for every number there is a "positive" and "negative" square root.
So if you want the square root of -4, you can do this: -4 = (-1)(4). So sqrt(-4) = sqrt[(-1)(4)] = sqrt(-1)*sqrt(4) = i*2 or -i*2. We usually write these as 2i and -2i.
How do you solve a quadratic equation with imaginary numbers?
You just plug in the coefficients, and do the normal operations. Of course you have to know how to calculate with complex numbers. Assuming the coefficients are real, you may at some moment get the root of a negative number. Say, for instance, you have the square root of minus 2, then the solution of that part is the square root of plus 2, multiplied by i.
If the original coefficients are complex, you may have to calculate the root of a complex number. This is a little more complicated. For this, you convert the complex number to polar coordinates - that is, to a length and an angle. Then, to actually take the square root, you take half the angle, and the square root of the distance - and convert back to rectangular coordinates (separating the real and the imaginary part). (For the second solution, add 180 degrees to the angle.)
Is the product of two imaginary numbers always an imaginary number?
If you are talking about pure imaginary numbers (a complex number with no real part) then no. Example: bi times ci where b and c are real numbers equals b*c*i² = b*c*(-1) = -b*c, which is a real number, because b & c & -1 are all real numbers. If you're talking about multiplying two complex numbers (a + bi)*(c + di), then the product will be complex, but it could be real or imaginary, depending on the values of a, b, c, & d.
Why are imaginary numbers important?
They are used for working out equations where the numbers you are working with are not physically possible, but we just imagine they are, such as the square root of a negative number
In engineering, especially Electrical Engineering, using complex numbers to represent signals (rather than sines and/or cosines) make comparing and working with signals easier.
What set of numbers does 30 belong to?
To any set that contains it!
It belongs to {30},
or {45, sqrt(2), 30, pi, -3/7},
or all whole numbers between 23 and 53,
or multiples of 5,
or square roots of 900,
or composite numbers,
or counting numbers,
or integers,
or rational numbers,
or real numbers,
or complex numbers,
etc.
What is William Rowan Hamilton's contribution to complex numbers?
Quaternions, which are an extension of complex numbers into 4 dimensions. See related link.
A complex number might not be a pure imaginary number?
True. Complex numbers have a real part and an imaginary part. If either one of these is zero, the complex number will be a pure real or a pure imaginary.
How do you convert a complex number from polar form into rectangular form?
If the polar coordinates of a complex number are (r,a) where r is the distance from the origin and a the angle made with the x axis, then the cartesian coordinates of the point are:
x = r*cos(a) and
y = r*sin(a)
What is the complex conjugate of 3i?
0+3i has a complex conjugate of 0-3i
thus when you multiply them together
(0+3i)(0-3i)= 0-9i2
i2= -1
0--9 = 0+9 =9 conjugates are used to eliminate the imaginary parts
When the sum of the complex numbers 3 plus 2I and 6 - 4I is graphed in which quadrant does it lie?
3+2i + 6-4i = 9-2i
The real part of this number is positive, therefore it lies in Q1 or Q4.
The imaginary part is negative, therefore it is in Q3 or Q4.
Q4 is the common possibility, therefore 9-2i is in Q4.
What is the imaginary number i to the 0 power equal to?
This is equal to 1. On the Wikipedia page for imaginary numbers, they have a table, but here is a summary for in:
n value of i^n
-- ------
-4, 1
-3, i
-2, -1
-1, -i
0, 1
1, i
2, -1
3, -i
4, 1
Notice there is a repeating pattern.
How do you write a complex number in standard form?
The standard from for a complex number is a + bi, where a and b can have any real value including zero and i = √-1
