Complex numbers are numbers that extend the concept of real numbers by introducing the imaginary unit, denoted as “i.” A complex number is typically expressed in the form a + bi, where “a” and “b” are real numbers. In this form, “a” represents the real part of the complex number, and “bi” represents the imaginary part.
The imaginary unit, “i,” is defined as the square root of -1, meaning that i2 = -1. This imaginary unit allows for the representation of numbers that cannot be expressed as real numbers alone. The real part of a complex number corresponds to the points on the real number line, while the imaginary part represents the points on the imaginary number line.
The conjugate of a complex number is obtained by changing the sign of its imaginary part while keeping the real part unchanged.
If we have a complex number z = a + bi, where “a” represents the real part and “b” represents the imaginary part, then the conjugate of z, denoted as z*, is given by z* = a – bi.
Real numbers are a broad class of numbers that include both rational and irrational numbers. They form a continuous and are used to represent quantities that can be measured on a continuous scale. Real numbers can be positive, negative, or zero.
The set of real numbers includes integers (such as -3, -2, -1, 0, 1, 2, 3), fractions or rational numbers (such as -1/2, 3/4, 5/2), and irrational numbers (such as √2, π, e). Rational numbers can be expressed as the ratio of two integers, while irrational numbers cannot be expressed as a simple fraction and have non-repeating decimal representations.