Mathematics is a very broad subject, and to make sense of it all we need to take a step back and understand the three different branches. This way, we can see how they are connected with each other. The first branch is mathematics as an art form that can be used for calculation and problem-solving. Arithmetic, algebra, geometry, and calculus comprise the second branch which uses mathematics in more analytical ways that focus on winning arguments between mathematicians.
Derivatives, analysis of functions, differential equations, etc., are included in this part of Mathematics because they represent ‘what is happening on our hands. The third branch includes, for example, the study of properties and relations between numbers, but more generally it is concerned with patterns that we can find in nature. Visit here, MP Board 12th Blueprint 2023
The definition of ‘pattern’ is broad, and so this branch of Mathematics also includes the study of animals, stars and galaxies, etc. In other words, this branch explains how things work in our universe. This last branch has to do with establishing ‘theorems’.The first two branches describe what we can do with mathematics and they are basically a vast range of tools that can be used to solve problems or prove our point when arguing with other mathematicians.
The third branch describes how things work in nature. The same thing that is true of ‘theorems’ in mathematics can be said about theorems in other branches of mathematics. For example, the theorem connecting integers and rational numbers is also a theorem in combinatorics which constructs another tool for solving problems. The only difference is that combinatorics does not use calculation. You can use study guides for ncert exemplar class 10 maths solutions
What is a theorem?
A ‘theorem‘ is a statement that has been proven to be true. When we say ‘proven to be true’, we mean using logical reasoning or a formal system. So theorems can be used as tools to prove other statements false. Theorems are part of Mathematics because they represent mathematicians’ way of thinking; they are their way of describing the rules that govern our universe, and they don’t just do that, they make sure we know it by proving what is true and what isn’t. So when mathematicians develop formal systems (logic), it allows them to create a structure that describes how things work in our universe. Infinity Learn will give the best guide in form of ncert exemplar class 10 maths pdf
Different mathematics theorem
1. It is not possible to have a square root of a negative number.
2. Adding 2 and -2 is equal to subtracting 5 from 8
3. The sum of the cubes of all numbers is equal to the cube of all numbers.
4. The sums of the reciprocals of any two integers are equal to each other
5 . The sum of two cubes is always greater than or equal to the sum of their squares.
6 . The whole is always greater than the sum of its parts
7. When you add to 5 all numbers that are multiples of 5, the difference remains the same total.
8. The smallest positive integer that has exactly 3 distinct divisors is itself.
9. If a number is a multiple of 3, then the sum of all its digits must be divisible by 3.
10. The square numbers are two multiplied together.
11. For any three natural numbers, the sum of the greatest and the smallest is always greater than or equal to twice their sum.
12 . Given that for any natural number ‘x’, dividing x + 1 by x – 1 gives a positive result, then there does not exist any prime number that divides it evenly into two equal parts.
13 . If a quadrilateral has all its vertices on the same circle then it is always a rectangle.
14 . The sum of the squares of the first ten natural numbers is approximately equal to 4.
15 . The square root of a negative number is impossible.
16 . Every whole number can be represented as a sum of at most nine integers.
17 . If the sum of two cubes is greater than either cube then it is possible to choose only one such cube that makes this inequality true.
18 . The Pythagorean theorem states that in any right triangle: the square of the hypotenuse equals the sum squared of the squares of the other two sides.
19. If a set of real numbers contains an infinite number of rational numbers then it contains infinitely many irrationals
20. If two sets A and B contain the same elements, then their product also contains the same elements.
21 . All prime numbers greater than 7 are odd.
22 . The sum of all even integers is equal to half of all natural numbers.
23 . The exact value of π is 3.141592653589793238462643383279…
24 . An equilateral triangle has the same area as any circle with the same radius.
25. Every natural number is the sum of two primes.
26 . All numbers are either positive or negative integers.
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Conclusion
An ‘axiom’ is a statement that is assumed to be true, but not proven.
In other words, an axiom never has a proof attached and it is never used as a tool in mathematics. They are part of all branches but they are included here because they represent assumptions that we have made in our thinking and they refer to Aristotle’s philosophy on logic.
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