properties of numbers Archives – Mathemafia http://sh016.global.temp.domains/~mathemaf/tag/properties-of-numbers/ Mathematics is like music - Enjoy it!!! Sat, 08 Mar 2025 11:37:52 +0000 en-US hourly 1 https://wordpress.org/?v=7.0 https://mathemafia.com/wp-content/uploads/2020/08/cropped-M__Logo__SM-1-32x32.png properties of numbers Archives – Mathemafia http://sh016.global.temp.domains/~mathemaf/tag/properties-of-numbers/ 32 32 Proof | Integers are infinitely many https://mathemafia.com/proof-integers-are-infinitely-many/ Tue, 22 Sep 2020 13:42:31 +0000 http://sh016.global.temp.domains/~mathemaf/?p=2085 Theorem: Prove that there are infinitely many integers. Proof is quite easy. You can watch video tutorial of proof as well.

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Proof - Integers are Infinitely Many

Theorem: Prove that there are infinitely many integers.

Proof: Let us assume that there are finitely many Integers.

\( \begin{aligned} &\Rightarrow \text{∃ a greatest integer, say } x \end{aligned}\)

Let us get a new number say y by adding 1 to x.

\( \begin{aligned} y&=x+1\end{aligned}\)
\( \begin{aligned} &Now \ \ \mathbf{y} \in\mathbb{Z} \\ &\text{because } \mathbf{x+1} \text{ is also an integer}\\\\&But \ \ y > x \end{aligned}\)

So we got another integer y > x.

This is a contradiction to our assumption that x is the greatest integer.

This means our assumption is wrong that there are finitely many integers.

Hence we can conclude that there are infinitely many integers.

Watch on YouTube

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]]> Mean method : Infinitely many rational numbers between any two rational numbers https://mathemafia.com/mean-method-infinitely-many-rational-numbers-between-any-two-rational-numbers/ Tue, 22 Sep 2020 13:36:47 +0000 http://sh016.global.temp.domains/~mathemaf/?p=2080 Concept: In this tutorial, you will learn that there are infinitely many rational numbers between any two rational numbers. There are multiple ways to learn it and I have taught by calculating the mean (average) of two numbers. Rational numbers…

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Concept: In this tutorial, you will learn that there are infinitely many rational numbers between any two rational numbers. There are multiple ways to learn it and I have taught by calculating the mean (average) of two numbers.

Rational numbers are infinitely many
Integers are infinitely many

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

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]]> Proof – Rational numbers are infinitely many https://mathemafia.com/rational-numbers-are-infinitely-many/ Tue, 18 Aug 2020 14:17:26 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1455 Theorem: Prove that there are infinitely many rational numbers. Proof is quite easy. You can watch video tutorial of proof as well.

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Rational Number - Definition

Before we prove that there are infinitely many rational numbers, let us recall the definition of Rational Numbers.

A number that is of the form p/q where p and q are integers and q≠0 is called a rational number.

\( \begin{aligned} \frac{p}{q}\ \ \ \ where \ \ p,q\in\mathbb{Z}, \ \ q\ne0 \end{aligned}\)

Proof - Rational Numbers are Infinitely Many

Theorem: Prove that there are infinitely many rational numbers.

Proof: Let us assume that there are finitely many Rational Numbers.

\( \begin{aligned} &\Rightarrow \text{∃ a greatest rational number, say } x\\\\ &\Rightarrow x = \frac{p}{q} \ \ where \ \ p,q\in\mathbb{Z}, \ \ q\ne0  \end{aligned}\)

Let us get a new number say y by adding 1 to x.

\( \begin{aligned} y&=x+1\\\\ &=\frac{p}{q}+1\\\\ &=\frac{p+q}{q} \end{aligned}\)
\( \begin{aligned} &Now \ \ p+q \in\mathbb{Z}\\\\ &\Rightarrow y = \frac{p+q}{q} \ \ \text{is also a rational number}\\\\ &Now \ \ y > x \end{aligned}\)

This is a contradiction that x is the greatest rational number.

This means our assumption is wrong that there are finitely many rational numbers.

Hence we can conclude that there are infinitely many rational numbers.

Watch on YouTube

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]]> Find a rational number between any two rational numbers https://mathemafia.com/find-a-rational-number-between-any-two-rational-numbers/ Tue, 18 Aug 2020 14:07:24 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1447 Concept: In this tutorial, you will learn to find a rational number between any two rational numbers by calculating the mean of the two numbers. It is quite simple method. There are more ways to find it, however, this is…

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Concept: In this tutorial, you will learn to find a rational number between any two rational numbers by calculating the mean of the two numbers. It is quite simple method. There are more ways to find it, however, this is the easiest method. The tutorial is in Hindi.

Commutative Law
Rational numbers are infinitely many

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

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]]> Commutative law explained for basic arithmetic operations https://mathemafia.com/commutative-law-explained-for-basic-arithmetic-operations/ Sun, 16 Aug 2020 07:04:54 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1390 Concept: In this tutorial, you will learn about the commutative property (commutative law) and its applicability on different arithmetic operations. You will learn that addition and multiplication hold the commutative property where as subtraction and division does not.  Identity, Inverse…

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Concept: In this tutorial, you will learn about the commutative property (commutative law) and its applicability on different arithmetic operations. You will learn that addition and multiplication hold the commutative property where as subtraction and division does not. 

Identity, Inverse
Rational number b/w two rational numbers

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

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]]> Additive Identity, Multiplicative identity, Additive inverse, Reciprocal https://mathemafia.com/additive-identity-multiplicative-identity-additive-inverse-reciprocal/ Sat, 15 Aug 2020 12:51:56 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1378 Concept: In this tutorial, you will learn the following: Additive Identity (0) Multiplicative identity (1) Additive inverse (-x) Reciprocal or Multiplicative Inverse (1/x where is not equal to zero) Closure property – Rational Numbers Commutative Law

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Concept: In this tutorial, you will learn the following:

  • Additive Identity (0)
  • Multiplicative identity (1)
  • Additive inverse (-x)
  • Reciprocal or Multiplicative Inverse (1/x where is not equal to zero)
Closure property - Rational Numbers
Commutative Law

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

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]]> Closure property – Rational Numbers https://mathemafia.com/closure-property-rational-numbers/ Sat, 15 Aug 2020 06:28:16 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1369 Concept: In this tutorial, you will learn about the closure property. It is discussed in detail for the following operations on rational numbers. Addition Subtraction Multiplication Division Closure property – Whole Numbers & Integers Identity & Inverse

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Concept: In this tutorial, you will learn about the closure property. It is discussed in detail for the following operations on rational numbers.

  • Addition
  • Subtraction
  • Multiplication
  • Division
Closure property - Whole Numbers & Integers
Identity & Inverse

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

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]]> Closure property – Whole Numbers and Integers https://mathemafia.com/closure-property-whole-numbers-and-integers/ Sat, 15 Aug 2020 06:27:54 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1368 Concept: In this tutorial, you will learn about the closure property. It is discussed in detail for the following operations on Whole numbers and Integers. Addition Subtraction Multiplication Division Closure property – Natural numbers Closure property – Rational numbers

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Concept: In this tutorial, you will learn about the closure property. It is discussed in detail for the following operations on Whole numbers and Integers.

  • Addition
  • Subtraction
  • Multiplication
  • Division
Closure property - Natural numbers
Closure property - Rational numbers

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

The post Closure property – Whole Numbers and Integers appeared first on Mathemafia.

]]> Closure property – Natural Numbers https://mathemafia.com/closure-property-natural-numbers/ Sat, 15 Aug 2020 06:18:15 +0000 http://sh016.global.temp.domains/~mathemaf/?p=1361 Concept: In this tutorial, you will learn about the closure property. It is discussed in detail for the following operations on natural numbers. Addition Subtraction Multiplication Division Associative property Closure property – Whole Numbers & Integers

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Concept: In this tutorial, you will learn about the closure property. It is discussed in detail for the following operations on natural numbers.

  • Addition
  • Subtraction
  • Multiplication
  • Division
Associative property
Closure property - Whole Numbers & Integers

Number System

Polynomials

Important Concepts – Miscellaneous

Surface area and Volumes

The post Closure property – Natural Numbers appeared first on Mathemafia.

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