Real Numbers vs Imaginary Numbers
Numbers are mathematical objects that are used to count and measure. The definition of it has changed over the years with the addition of zero, negative numbers, rational numbers, irrational numbers and imaginary numbers. Even though the abstract foundation of number systems relates to algebraic structures such as groups, rings and fields, only an intuitive idea is presented here.
What is a real number?
Informally defining, a real number is a number whose square is non-negative. In mathematical notation, we denote the set of real numbers by the symbol R. Therefore for all x, if x ϵ R then x2 ≥ 0. In a more rigorous way, can introduce the set of real numbers as the unique, complete totally ordered field with the binary operation + and . along with the order relation <. This order relation follows the trichotomy law, which states that given two real numbers x and y, one and only one of these 3 holds; x >y, x <y or x =y.
A real number can be either algebraic or transcendental depending on whether it is a root of a polynomial equation with integer coefficients or not. Also, a real number can be either rational or irrational depending on whether it can be expressed as a ratio of two integers or not. For example, 2.5 is a real number, which is algebraic and rational, but ᴫ is irrational as well as transcendental.
The set of real numbers is complete. It means that for each nonempty subset of real numbers that is bounded above, has a least upper bound, and from this, it can be deduced that for each nonempty subset of real numbers that is bounded below, has the greatest lower bound. This distinguishes the set of real numbers from the set of rational numbers. One can argue that the set of real numbers is built by filling the gaps of set of incomplete rational numbers, the gaps being irrational numbers.
What is an imaginary number?
An imaginary number is a number whose square is negative. In other words, numbers like √(-1), √(-100) and √(-e) are imaginary numbers. All the imaginary numbers can be written in the form a i where i is the ‘imaginary unit’ √(-1) and a is a non-zero real number. (Observe that i2 = -1). Though these numbers seem to be non-real and as the name suggests non-existent, they are used in many essential real world applications, in fields like aviation, electronics and engineering.
What is the difference between real numbers and imaginary numbers?
• The square of a real number is non-negative, but the square of an imaginary number is negative.
• Set of real numbers forms a complete totally ordered field whereas the set of imaginary numbers is neither complete nor ordered.