Mathematical principles are founded on the defining boundaries we impose on the numerical values under consideration. When we state that 5 + 3 = 8 or 3 x 5 = 15, we are referring to integer amounts, which are the same kind of numbers that we all learned to count in school. What most people believe to be self-evident mathematical rules true at all times but for any purpose, it depends on how we define a number.
When calculating numbers in AC circuits (alternating current), for example, we discover that the “real” number quantities that served us so well in DC circuit analysis are insufficient for the purpose of expressing AC numbers.
We know that voltages accumulate when linked in series, but we also know that connecting a 5-volt AC source in series with a 6-volt AC source results in a total voltage of 9 volts. Is this a violation of the unchallengeable principles of mathematics?
No, it simply implies that the principles of “real” numbers do not apply to the kind of quantities observed in alternating current circuits, where each variable has both a point and a scale.
As a result, we must employ a new type of numerical quantity, or object, for AC circuits, i.e., complex numbers rather than real numbers. This alternative system of numbers comes with a distinct set of rules describing how they interact with one another. Within the scope and meaning of real numbers, an equation like “5 + 6 = 9” is gibberish, yet it fits neatly within the scope and meaning of complex numbers.
For example, a right triangle with adjacent and opposite sides of 5 and 6 having a hypotenuse of 9. Single-dimension real numbers cannot be added trigonometrically to one another, but complex numbers can be added because they are two-dimensional.
What are Logics?
Logic is similar to mathematics but it depends if laws of logic are determined by how we define what a proposition is. Aristotle, a Greek philosopher, developed a logic system based on only two sorts of propositions: true and false.
His bivalent concept of truth resulted in the four core principles of logic.
- The Law of Identity (A is A)
- The Law of Rational Inference
- The Law of the Excluded Middle (A or non-A)
- The Law of Non-contradiction (A is not non-A)
These rules and Laws operate within the framework of logic when a proposition may only have one of two potential values, but they may not apply when propositions can have values other than “true” or “false.”
In reality, substantial research has been conducted and continues to be conducted on, uncertain or multiple valued logic, in which propositions can be true or untrue to varying extents. Because they are based on the premise of bivalence, “Laws” such as the Law of the Excluded Middle simply do not apply in such a logical system.
Similarly, many principles that would violate Aristotelian logic’s Law of Non-contradiction have validity in uncertain logic. Again, the Laws specifying their functions and interactions are determined by the defining bounds of propositional values.
How Boolean Algebra came into existence?
The English mathematician George Boole attempted to give Aristotle’s reasoning or logical theory a symbolic form.
In 1854, Boole published An Investigation of the Laws of Thought, on Which Are Based the Mathematical Theories of Logic and Probabilities, which formalized various principles of interaction between mathematical quantities that could only have one of two potential values: true or untrue, 1 or 0.
Boolean algebra is the name given to the mathematical theory of George Boole. All mathematical operations with Boolean numbers may only have one of two outcomes: 0 or 1. In the Boolean universe, there is no such thing as “3,” “-2,”, “0.3,” or a fraction. It is a universe in which all other numbers are rendered null and void by decree.
As one might expect, this is not the type of arithmetic you apply while calculating voltage via a circuit or tallying financial ledgers.
An American Mathematician Claude Shannon understood how Boolean algebra might be used to on-and-off circuits, where all signals are classified as either “low” (0) or “high” (1).
His thesis “A Symbolic Analysis of Relay and Switching Circuits” published in 1938 used Boole’s theoretical work in ways that Boole could never have planned, providing us with a strong mathematical tool for creating and evaluating digital circuits. Several tools such as Boolean Algebra Calculator, are based on those Laws and symbolic analysis.
Difference between Boolean Algebra and Normal Algebra
There are many resemblances between normal and Boolean algebra, which involves so-called real numbers which you will discover in this section.
Just keep in mind that the number system that defines Boolean algebra is extremely limited in scope and that each Boolean variable may only have one of two potential values which are 0 or 1.
As a result, the rules of Boolean algebra frequently diverge from the rules of real-number or normal algebra, allowing claims like 2 + 2 = 2 to be made that would ordinarily be regarded as ridiculous.
The silly talk of Boolean algebra vanishes if you understand the proposition that all quantities in Boolean algebra are confined to the two options of 0 and 1, as well as the wider philosophical concept of Laws dependent on quantitative definitions.
Are Binary Numbers and Boolean Numbers the same?
No, they are not the same. It is important to understand that Boolean numbers are not comparable to binary numbers. Binary numbers are only another notation for real numbers, however, Boolean numbers represent an altogether distinct mathematical system as compared to real numbers.
Because both binary notation and Boolean employ the same two symbols 0 and 1, they are frequently mistaken. The distinction is that Boolean values are limited to a single bit either 0 or 1, but binary numbers can be made of several bits adding up to a value of any finite size in point-subjective form.
Introduction to Boolean algebra went much deep and I couldn’t stop writing because I wanted to explore and share several aspects of it with you. In short, Boolean algebra is pretty much different from normal algebra and Boolean numbers should not be mistaken as binary numbers. I will explore applications of Boolean algebra in another post. Take care of yourself and don’t forget to share your thoughts in the comment section.