Yes. This is easy to prove; in the following, I'll use "^" for powers. Let the complex number be (a + bi), then its conjugate, by definition, is (a - bi). Multiplying them, you get a^2 + abi - abi + bi^2 = a^2 + bi^2 = a^2 - b^2 (since i^2 = -1).Update: One interesting, and quite useful, property is that the product of the complex number and its conjugate is equal to the square of the absolute value of the complex number.
if z = x + iy (x and y are both real) then its conjuguate is x - iy, their product is
( x + iy ) ( x - iy ) = x^2 -xiy + xiy - i^2 y ^2 = x^2 + y ^2, then it's a real number
Yes.
Complex numbers form: a + bi where a and b are real numbers. The conjugate of a + bi is a - bi If you multiply a complex number by its conjugate, the product will be a real number, such as (a + bi)(a - bi) = a2 - (bi)2 = a2 - b2i2 = a2 - b2(-1) = a2 + b2
Not necessarily. It can be wholly imaginary.For example, 1 + i actually has two complex conjugates. Most schools will teach you that the complex conjugate is 1 - i. However, -1 + i is also a conjugate for 1 + i. (Their product is -1 times the product of the "normal" conjugate pair).The sum of 1 + i and -1 + i = 2i
The conjugate will have equal magnitude. The angle from the real axis will be the same angle measure (but opposite direction).
For a complex number (a + bi), its conjugate is (a - bi). If the number is graphically plotted on the Complex Plane as [a,b], where the Real number is the horizontal component and Imaginary is vertical component, the Complex Conjugate is the point which is reflected across the real (horizontal) axis.
Their sum is real.
When a complex number is multiplied by its conjugate, the product is a real number and the imaginary number disappears.
Graphically, the conjugate of a complex number is its reflection on the real axis.
Complex numbers form: a + bi where a and b are real numbers. The conjugate of a + bi is a - bi If you multiply a complex number by its conjugate, the product will be a real number, such as (a + bi)(a - bi) = a2 - (bi)2 = a2 - b2i2 = a2 - b2(-1) = a2 + b2
Assuming that the question is in the context of complex number, the product of any real number with itself (its square) is a real number.
Assuming that the question is in the context of complex number, the product of any real number with itself (its square) is a real number.
Not necessarily. It can be wholly imaginary.For example, 1 + i actually has two complex conjugates. Most schools will teach you that the complex conjugate is 1 - i. However, -1 + i is also a conjugate for 1 + i. (Their product is -1 times the product of the "normal" conjugate pair).The sum of 1 + i and -1 + i = 2i
Since the imaginary portion of a real number is zero, the complex conjugate of a real number is the same number.
The conjugate will have equal magnitude. The angle from the real axis will be the same angle measure (but opposite direction).
For a complex number (a + bi), its conjugate is (a - bi). If the number is graphically plotted on the Complex Plane as [a,b], where the Real number is the horizontal component and Imaginary is vertical component, the Complex Conjugate is the point which is reflected across the real (horizontal) axis.
Their sum is real.
A number multiplied by its complex conjugate will result in a real number. Also, adding a number to its conjugate will result in a real number. But typically the multiplication is what is used.
Aamir jamal; All real numbers are complex numbers with 0 as its imaginary part.A real number is self-conjugate. e.g;a+0i self conjugate =a-0i i=iota