Introduction to Euclid's Geometry

Axioms, Conjectures and Theorems

Axioms or Postulate is defined as a statement that is accepted as true and correct, called as a theorem in mathematics. Axioms present itself as self-evident on which you can base any arguments or inference. These are universally accepted and general truth. 0 is a natural number, is an example of axiom. What are Axiom, Theory and a Conjecture? Let us explore these concepts in detail and begin to understand how things work in Mathematics!

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Mathematical Statements

In Mathematics, a statement is something that can either be true or false for everyone. For example, The mass of Earth is greater than the Moon or the sun rises in the East. In other words, if a statement has the same meaning everywhere and can either be true or false, it is a Mathematical statement.

A statement is a non-mathematical statement if it does not have a fixed meaning, or in other words, is an ambiguous statement. For example, “Computers are good and easy”. The statement is an opinion and will have a different meaning for different people, so its meaning is ambiguous. As another example, a statement like “close the door” is also not a mathematical statement. It doesn’t have a true or a false value. Therefore it is not a Mathematical Statement.

Now the question is how do we know which statement is true and which is false? Let us look at an example. Rahul, a student goes out to buy ice-cream for his friends one evening. He has 2 friends. If all 3 of them (including Rahul) want 1 ice cream each, how many ice-creams should Rahul buy? Silly, you may say, as obviously, Rahul needs to buy 3 ice creams for all 3 of them to have one ice-cream each. Three ice-creams is the correct answer but can you prove that it is the answer?

In the above example, we counted the number of students and equated that number to the number of ice-creams. We followed a logical path. Also, we assumed that every student will get exactly one ice-cream. All of these are an example of a mathematical statement!

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Axiom

The word ‘Axiom’ is derived from the Greek word ‘Axioma’ meaning ‘true without needing a proof’. A mathematical statement which we assume to be true without a proof is called an axiom. Therefore, they are statements that are standalone and indisputable in their origins. In simpler words, these are truths that form the basis for all other derivations and have been derived from the basis of everyday experiences. In addition to this, there is no evidence opposing them.Axioms

Examples of Axioms

Examples of axioms can be 2+2=4, 3 x 3=4 etc. In geometry, we have a similar statement that a line can extend to infinity. This is an Axiom because you do not need a proof to state its truth as it is evident in itself.

Conjecture

A conjecture is such a mathematical statement whose truth or falsity we don’t know yet. In other words, a statement that you believe to be true but have not proved to be true is called a conjecture. For example: What is the next number in the series 3  6  9 12? The answer is ’15’. On careful observation, we see that each succeeding number is greater than the previous one by a difference of ‘3’. So even though we don’t see the next number we can correctly guess it by observing the pattern generated.

So, one of the conjectures is that “the next number is 15”. Another conjecture could be “the next number is (15 × 1) + 0”. None of the two has a proof but both follow from simple mathematical rules or axioms.

Conjectures play a very important role in problem-solving in Mathematics and Geometry, where the solution is not always apparent and we generate the solution by following a series of steps. Generally, each of these steps is a ‘Conjecture’ over the previous step.

Theorem

A mathematical statement that we know is true and which has a proof is a theorem. We can further explain it as a series of Conjectures (proof) that combine together to give a true result. So if a statement is always true and doesn’t need proof, it is an axiom. If it needs a proof, it is a conjecture. A statement that has been proven by logical arguments based on axioms, is a theorem. We generate a theorem by the way of analysis and proof. Consider them your weapons, Superheroes as you start on this journey through the mazes of mathematics and arrive at solutions and save the day.

Axiom

Solved Examples For You

Solved Examples For You

Question 1: The formation or expression of an opinion or theory without sufficient evidence for proof is known as

A) Axioms                 B) Conjecture             C) Corollary               D) Theorem

Answer : B) Conjecture. As we discussed above, a conjecture is a statement that we arrive on with logical reasoning in mathematics, it is something that has not been proved yet. So we write the answer as B) conjecture.

Question 2: What is a true axiom?

Answer: An axiom refers to a statement which everybody believes to be true, such as “the only constant changes.” Mathematicians make use of the word axiom to refer to conventional proof. The term axiom is derived from a Greek word that means “worthy.” Thus, an axiom refers to a worthy, established fact.

Question 3: What is the difference between Axioms and Postulates?

Answer: The difference between these two is that an axiom usually is true for any field in science, whereas a postulate may be explicit on a particular field. Moreover, it is impossible to prove from other axioms, whereas postulates are provable to axioms.

Question 4: How many axioms are there?

Answer: There are five axioms. As you know it is a mathematical statement which we assume to be true. Thus, the five basic axioms of algebra are the reflexive axiom, symmetric axiom, transitive axiom, additive axiom and multiplicative axiom.

Question 5: Why are axioms important?

Answer: It is quite essential to get axioms right, as all of the mathematics depends on them. In other words, if there are too few axioms, you can prove very little and thus math will not be that interesting. Similarly, if there are too many axioms, you will be able to prove almost anything, and it will also make mathematics interesting.

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Subha
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Subha

I believe that natural number begin from 1,2,3….. and 0 is a whole number. 0 is not natural number

1984
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1984

3 x 4 is not 4

Peters
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never trust a source without an author i guess

Payam
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Payam

The last sent. should be changed into “it will also make mathematics NOT interesting.”

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