Learn Basic Postulates in Geometry with Our Experts

The way I view Euclid's postulates are as follows:

A line segment can be made between any two points on surface A.

A line segment can be continued in its direction infinitely on surface A.

Any line segment can form the diameter of a circle on surface A.

The result of an isometry upon a figure containing a right angle preserves the right angle as a right angle on surface A.

If a straight line falling on two straight lines make the interior angles on the same side less than 180 degrees in total, the two straight lines will eventually intersect on the side where the sum of the angles is less than l80 degrees.

The way I view the five postulates is simple. Each postulate defines some quality a surface has.

For instance, 2 seems to define whether a surface is infinite/looped or finite/bounded, 3 seems to force a surface to be circular (or a union of circular subsets), and 5 I believe change the constant curvature of a surface (wether it is 0 or nonzero).

I want to determine what "quality" 1 and 4 define in the context of the surface itself. 1 seems to imply discontinuity vs continuity, and I think 4 would imply non-constant curvature. However, I am unsure. Ultimately I would like to assign each of these a quality of a surface that they define such that all surfaces can be "categorized" under some combination of postulates, but that is irrelevant.

I am merely asking:

What two surfaces individually violate the first postulate and violate the fourth postulate?

In his 5 postulates Euclid said:"All the right angles are equal". What he really meant by that ? Does he meant that we can impose any right angle on each other or the meaning there was that if you find this angle through the length of arc and radius of the circle then you get the same number ? I also heard that this postulate can be proven (and I remembered that I saw such a question there, but now I can't find it). So what is the proof (based on the definition of a right angle like right angle is an angle which is equal to its adjacent angle)

Is the proof of Fermat's last theorem solely based on the Peano's postulates + first order logic? Or it contains other axiomatic systems as well? What does it mean from foundations of math perspective to use several axiomatic systems to prove a conjecture? Do we know these axiomatic systems are consistent with one another? I'm not sure if I am asking it the right way, but I think logicians only prove the consistency of the axioms of one system not two different systems.

Intuitionism refuses the Cantor hypothesis about continuum as a hypothesis meainingless from the intuitionistic point of view Also, 'the tertium non datur' principle:
$A\vee <!--\; \vee \; -->\mathrm{\neg <!--\; \neg \; -->}A$
is rejected 'a priori' in the sense that we can only prove the validity of $A$ or $\mathrm{\neg <!--\; \neg \; -->}A$. My question is: given the rejection of these principles, in particular the second, is it still possible to keep the validity of the mathematical analysis within the intuitionism? Or maybe, accepting the intuitionistic approach, you need to develop a new kind of analysis, because, for example, it's impossible to use the 'reductio ad absurdum' to prove a theorem? Thanks

I hear about axioms in set theory and postulates in geometry, but they seem like the same thing. Do they mean the same thing but then are used in different instances or what? Is one word more applicable in one case more than the other? I never hear of axioms in geometry or postulates in set theory. Are axioms more formal and postulates used more informally?

I want to show that $2p_{n-<!--\; -\; -->2}\ge <!--\; \ge \; -->p_n-<!--\; -\; -->1$...
Bertand's postulate shows us that $4p_{n-<!--\; -\; -->2}\ge <!--\; \ge \; -->p_n$ but can we improve on this?
any ideas?

Hi guys I just need to know if my answer is right.

The question is

1) Euclids $4^{th}$ postulate is "That all right angles are equal to one another". Why is this not obvious?

My answer:

When I read this question I am like it is obvious, so I got kind of confused. But I took a crack at it anyways.

If you read the postulate. This is not obvious because you dont know if the right angle is a right angle. What if a triangle was drawn differently but with one line perpendicular to another. You need to know if the line is perpendicular or not, and that is why it is not obvious.

Could someone tell me if that is right or should I add in more info.

I don't find Postulate 13 of Spivak's Calculus trivial, nor can I understand why it's true.
Postulate 13: Every non-empty set of real numbers that is bounded above has a least upper bound (sup).
Why is this postulate true? Any proof/intuition behind it?

Edit: Let me pose a few questions.
1. Suppose I decide to devise a pathological function $\mathbb{R}$$\to <!--\; \to \; -->$$\mathbb{R}$ which is bounded above but has no supremum. Why will I fail to find such $f$? If you simply take it as an axiom, there's no guarantee I won't be successful.
2. Suppose $S$ is an arbitrary non-empty set of real numbers that is bounded above. Does there exist an algorithm to determine $sup(S)$?

Bertrand's postulate

claim :

for all $5<n\in <!--\; \in \; -->\mathbb{N}$ there is at least two prime numbers different in the opening interval $(n,2n)$.Use reinforcement of Bertrand's postulate in order to prove that :
for all $10<n\in <!--\; \in \; -->\mathbb{N}$ Exists that has at least two Prime factors in the factorization of $n!$ which appear with a power of 1.
Example : $n=11$ the primes 7,11 are such prime's meet the conditions.
But , for $n=10$ it dosen't meet the conditions because only 7 appears with power 1 in the factorization of 10!.

Hint : Consider two cases, $n$ even or $n$ odd

Attempt:
we need to use the claim in order to Implement the solution

Case (1): $n$ even
if $n$ is even we can rewrite $n=2t$
if we use the claim we can get $(2t,4t)$

Case (2): n odd
if $n$ is odd we can rewrite $n=2t+1$
if we use the claim we can get $(2t+1,2(2t+1))$

I came up with this question- how would you show 4 not equal to 6 (or m not equal to m+n ( n not 0)), using only Peano's Postulates?

I can see a number of things go wrong- for instance the Principle of Mathematical Induction seems to fail. Also possibly 0 seems to be in the image of the successor function in that case.

Let $A$,$B$ be two points on opposite sides of a line $l$. Then the line segment $AB$ intersects $l$.

My question is:
1. Using only the 4 postulates of Euclid, is there a way to make precise the meaning of ``opposite sides"?
2. Is the intersection guaranteed to exist using only the 4 postulates? If so, why is it true?

Suppose I have this boolean expression:
W'XYZ + WX'YZ + WXY'Z + WXYZ' + WXYZ
How would I go about simplifying this without using a K-map? Using K-map, the simplified form is XYZ + WXY + WXZ + WYZ. I read about the redundancy theorem somewhere, would rather not use that as well.

Using Bertrand's postulate which states:
For every integer $n\ge <!--\; \ge \; -->1$ there is a prime number p such that $n<p\le <!--\; \le \; -->2n$
Prove that there exists infinitely many primes whose decimal expansion starts with 1.
I'm guessing I need to use something equal to 10 in this as it is a decimal expansion, but I'm not seeing where to start?
Any guidance would be great, thank you

How can I prove the following two questions:

Prove using Peano's Postulates for the Natural Numbers that if a and b are two natural numbers such that a + b = a, then b must be 0?

Prove using Peano's Postulates for the Natural Numbers that if a and b are natural numbers then: a + b = 0 if and only if a = 0 and b = 0?

I understand the basic postulates, but not sure how to apply them to these specific questions. The questions seem so basic and obvious, but when it comes to applying the postulates I am lost.

This may be a dumb question.

The Poisson postulates are:

1. $P(n=1,h)=\mathrm{\lambda <!--\; \lambda \; -->}h+o(h)$
2. $\underset{i=2}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}P(n=i,h)=o(h)$
3. Events in nonoverlapping intervals are independent

What ensures that $\mathrm{\lambda <!--\; \lambda \; -->}h\in <!--\; \in \; -->[0,1]$ irrespective of the value of $\mathrm{\lambda <!--\; \lambda \; -->}$ ?