Best Answer

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.

Q: Is The product of a complex number and its conjugate is a real number.?

Write your answer...

Submit

Still have questions?

Continue Learning about Math & Arithmetic

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.

Related questions

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.

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.

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

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).

The graph of a complex number and its conjugate in the complex plane are reflections of each other across the real axis. If a complex number is represented as z = a + bi, its conjugate z* is a - bi. This symmetry across the real axis is a property of the complex conjugate relationship.

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.