Introduction to Euclid's Geometry — Question 7
Back to all questionsAxiom 5 states that :
'Whole is always greater than its part'. This is a 'universal truth' because it holds true in any field of mathematics and in other disciplinarians of science as well.
Let us take two cases: one in the field of mathematics and one other than that.
Case 1 — Let t represent a whole quantity and only a, b, c are parts of it.
Such that :
t = a + b + c
Clearly, t will be greater than all of its parts a, b and c. Therefore, it is rightly said that the whole is greater than the part.
Case 2 — Let us consider continent Asia. Then, let us consider a country India which belongs to Asia. India is a part of Asia and it can also be observed that Asia is greater than India. That is why we can say that the whole is greater than the part.
Euclid’s Geometry — Interactive Study Guide
The Big Idea
Euclid started with things everyone agrees on (axioms and postulates) and built all of geometry by logical deduction. This is the axiomatic method — start from accepted truths, derive everything else.
Axiom vs Postulate vs Theorem
Axiom: Self-evident truth used everywhere in mathematics.
Postulate: Assumption specific to geometry (cannot be proved, must be accepted).
Theorem: A statement that has been proved using axioms, postulates, and logic.
The Famous 5th Postulate
Euclid’s 5th postulate is about parallel lines. For over 2000 years, mathematicians tried to prove it from the other 4 postulates. They failed — because it’s independent! Changing the 5th postulate leads to non-Euclidean geometry.
Quick Self-Check
- State Euclid’s first axiom. (Things equal to the same thing are equal to one another.)
- How many lines can be drawn through two distinct points? (Exactly one — Postulate 1)
- What is the modern equivalent of Euclid’s 5th postulate? (Playfair’s axiom)