Conjugate of a Complex Number
The conjugate of a complex number is another complex number that has the same real part as the original complex number, and the magnitude of the imaginary part is also the same but with the opposite sign. Two complex numbers are said to be each other’s conjugates if their sum and product are real numbers.
- The conjugate of the complex number z = a + ib is z̅ = a − ib.
- Sum of a complex number and its conjugate = z + z̅ = (a + ib) + (a − ib) = 2a
- Product of a complex number and its conjugate = z × z̅ = (a + ib)×(a − ib)= a2+b2
Is Every Real Number a Complex Number?
A complex number is referred to as the sum of a real number and an imaginary number. It is generally expressed as “z” and is written in the form of a + ib, where a and b are real numbers and i = √(-1). Here, “a” is a real part that is represented as Re(z) and “ib” is an imaginary part that is represented as Im(z). Some examples of complex numbers are 2 + 3i, 5–7i, 3 + i√4, etc. The imaginary number is generally expressed either as “i” or “j”, whose value is equal to √(-1). Hence, the square of an imaginary number gives us a negative value. The square root of negative numbers can be calculated using complex numbers. Some applications of complex numbers are in signal processing, fluid dynamics, quantum mechanics, electromagnetism, vibration analysis, and also many scientific research areas.
Real numbers are referred to as the union of the set of rational numbers and the set of irrational numbers, i.e., positive numbers, whole numbers, integers, rational numbers, irrational numbers, etc. are real numbers. Some examples of real numbers are -4, -7/11, 0, 9, √6, 3.8, etc.
A number that gives a negative value when squared is called an imaginary number. It is the product of a non-zero real number and the imaginary unit “i”, whose value is √(-1). An imaginary number can also be defined as the square root of negative numbers. Some examples of imaginary numbers are -2i, √5i, 3i, etc.