Oh, dude, you're hitting me with some math vibes! So, like, a complex number is in the form a + bi, right? The conjugate of that is a - bi. If you square a + bi, you get (a^2 - b^2) + 2abi. The conjugate of that is (a^2 - b^2) - 2abi. So, for a complex number to be equal to the square of its conjugate, you'd need (a^2 - b^2) + 2abi = (a^2 - b^2) - 2abi, which means b has to be 0. So, the complex number would be a real number.
the absolute value of x + iy is equal to (x^2+y^2)^.5 and is the same for the conjugate, x-iy
The square root
If you understand what the absolute value of a complex number is, skip to the tl;dr part at the bottom. The absolute value can be thought of as a sorts of 'norm', because it assigns a positive value to a number, which represents that number's "distance" from zero (except for the number zero, which has an absolute value of zero). For real numbers, the "distance" from zero is merely the number without it's sign. For complex numbers, the "distance" from zero is the length of the line drawn from 0 to the number plotted on the complex plane. In order to see why, take any complex number of the form a + b*i, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. In order to plot this number on a complex plane, just simply draw a normal graph. The number is located at (a,b). In order to determine the distance from 0 (0,0) to our number (a,b) we draw a triangle using these three points: (0,0) (a,0) (a,b) Where the points (0,0) and (a,b) form the hypotenuse. The length of the hypotenuse is also the "distance" of a + b*i from zero. Because the legs run parallel to the x and y axes, the lengths of the two legs are 'a' and 'b'. By using the Pythagorean theorem, we can find the length of the hypotenuse as (a2 + b2)(1/2). Because the length of the hypotenuse is also the 'distance' of the complex number from zero on the complex plane, we have the definition: |a + b*i| = (a2 + b2)(1/2) ALRIGHT, almost there. tl;dr: Remember that the complex conjugate of a complex number a + b*i is a + (-b)*i. By plugging this into the Pythagorean theorem, we have: b2 = (-b)2 So: (a2 + (-b)2)(1/2) = (a2 + b2)(1/2) QED.
No, because 1 times itself is one, making it not a square number. It has to equal a different number than the number times itself. * * * * * A totally incorrect answer - on two counts. (a) a square number does not have to be different and, (b) even if that were the case, 1 is the square of -1 and -1 is not the same as 1.
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).
Yes they do, complex conjugate only flips the sign of the imaginary part.
yes
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A conjugate number refers to a complex number having both the imaginary and real parts of opposite signs and equal magnitude.
The complex conjugate pair, 2 - 5.2915i and 2 + 5.2915i, approx, where i is the imaginary square root of -1.
The complex conjugate pair, -125 - 233.1845*i and -125 + 233.1845*i where i is the imaginary square root of -1.
The conjugate will have equal magnitude. The angle from the real axis will be the same angle measure (but opposite direction).
Hermitian matrix (please note spelling): a square matrix with complex elements that is equal to its conjugate transpose.
The complex conjugate pair, 2.5 ± 1.9365i where iis the imaginary square root of -1.
The complex conjugate pair, 0.5±3.4278i where i is the imaginary square root of -1.
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