How do you prove two complex numbers are equal?

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. I.e., a+bi = c+di if and only if a = c, and b = d.

How do you prove that complex numbers are real?

In fact, one of the most helpful aspects of the complex conjugate is to test if a complex number z = a + bi is real. A complex number is real if and only if z = a +0i; in other words, a complex number is real if it has an imaginary part of 0.

What are nonzero complex numbers?

A quantity which does not equal zero is said to be nonzero. A real nonzero number must be either positive or negative, and a complex nonzero number can have either real or imaginary part nonzero.

What does z equal in complex numbers?

We often use the variable z=a+bi to represent a complex number. The number a is called the real part of z: Re z while b is called the imaginary part of z: Im z. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

Is z 0 a complex number?

Complex Numbers. When Re(z) = 0 we say that z is pure imaginary; when Im(z) = 0 we say that z is pure real. Both Re(z) and Im(z) are real numbers. Thus, any complex number can be pictured as an ordered pair of real numbers, (a, b) .

How do you find the z of a complex number?

You can easily check that a complex number z = x + yi times its conjugate x – yi is the square of its absolute value |z|2. Therefore, 1/z is the conjugate of z divided by the square of its absolute value |z|2.

What is nonzero rational number example?

A non-zero rational number includes integers, fractions, square roots and π that are not 0 and are not square roots of any negative numbers. Also, if x and y are non-zero rational numbers, then x.y is also a non-zero rational number.

Does the triangle inequality work for complex numbers?

The inequality Iwl + Izi ~ Iw + zl is called the Triangle Inequality for complex numbers. Given the name, you might think the inequality has something to do with geometry. You’re right; using a geometric representation of complex numbers and complex addition, we can prove the Triangle Inequality quite easily.