Even Numbers

Introduction

Even numbers are numbers in the form of n = 2k, which means the number which can be divided by 2. Mathematics is a branch of science that arose due to the needs of man for calculation in business, knowing the relationship between numbers, measuring the land, in predicting cosmic events.

Mathematics is used as an important tool in many fields of the world such as natural sciences, engineering, medicine, finance, and social sciences. Applied mathematics deals with the use of mathematics in other fields to stimulate and apply new scientific discoveries. Mathematics is not only numbers and their procedures (addition, subtraction, multiplication, division), as well as shapes (shapes), but it is a scientific tool that is constantly growing with scientific research and its applications.

Numbers

• There are many different types of numbers in mathematics.Natural numbers (1, 2, 3, 4, . . . ) are the basic examples of numbers.

• The notation used to represent numbers is called numerical. It can be assumed that the concept of number was created when man learned to count his fingers at the time of his birth. The development of the concept of numbers is the origin of mathematics.

Number System

Natural Numbers and Whole numbers

• The most familiar numbers in practice, the numbers used for counting, are the natural numbers

• A natural number is defined as a sequence of positive integers (1, 2, 3, 4, . . . ) and whole numbers as a sequence of non-negative integers (0, 1, 2, 3, 4, . . . ).

Integers

• Integers denote the positive natural numbers (1, 2, 3, 4, . . . ) their negatives (−1, −2, −3, −4, . . . ) and the zero decimal. For example, 13, 9, and −1204 are integers.

• Adding positive and negative numbers to natural numbers forms the integer.

• Z is the symbol used to denote the set of integers.

Rational Numbers

• Fractional numbers or rational numbers that are represented by whole numbers above and below a fraction represented by whole number ratios such as half, quarter, and one-third are called rational numbers

• This category of numbers is denoted by the letter Q.

Real Numbers

• It was not until the 19th century that man knew without a doubt that there was a single supernumerary set {R} that included the rational number Q and contained all the numbers people encounter in life.

• The elements of the sequence {R} are called real numbers. In this, if the rational numbers are removed, the remaining numbers are called non-rational numbers

• There are two types of irrational numbers, algebraic numbers and transcendental numbers.

Even Numbers

• Even numbers are numbers that do not leave a remainder when divided by two. Multiples of two are even numbers.

• Even numbers are numbers whose last digit can be 0, 2, 4, 6 𝑜𝑟 8.

• For example, 0,2,4,6,8,10,12,14,16,18 .......

Properties of even numbers

Addition Property	Subtraction Property	Multiplication property
Even+Even=Even	Even−Even=Even	Even×=Even=Even
Even+Odd=Odd	Even−Odd=Odd	Even×Odd=Even
Odd+Even=Odd	Odd−Even=Odd	Odd×Even=Even
Odd+Odd=Even	Odd−Odd=Even	Odd×Odd=Odd

• The addition or subtraction of two even numbers or odd numbers will result in an even number.

• The multiplication of any even number with any odd or even numbers will result in an even number

The only even prime: 2

• A prime number is a natural number greater than 1 that has no positive divisors other than 1 and the same number.

• 1 and other natural numbers with denominators other than the same number (except 1) are called composite numbers. For example, the natural number 11 is a prime number. It has no divisors other than 1.

• 2 is the only even prime number because the divisor of 2 is only 1 and 2. It cannot be divided by any other number. 2 is the smallest even number.

Algebra of even Numbers

The even numbers are also written as $\mathrm{x\:=\:2n}$. Because any number multiplied by 2 results in an even number. Here n is the positive integer. But the odd number is written as $\mathrm{x\:=\:2n\:+\:1}$.If $\mathrm{n\:=\:3\:,\:x\:=\:2\times\:3\:=\:6}$. 6 is an even number.

Solved Examples

1)If the sum of two numbers is 𝟓𝟓 and their difference is 𝟕, find the numbers. Also, find which is an even number in those numbers

Ans

Let the two numbers be 𝑥 and 𝑦. Here $\mathrm{x\:>\:y}$

From the given instructions,

$$\mathrm{x\:+\:y\:=\:55\:\:\:\:\rightarrow\:(1)}$$

$$\mathrm{x\:-\:y\:=\:7\:\:\:\:\rightarrow\:(2)}$$

$$\mathrm{x\:=\:7\:+\:y\:\:\:\:\rightarrow\:(3)}$$

Substitute the value of 𝑥 in equation (1)

$$\mathrm{7\:+\:y\:+\:=\:55}$$

$$\mathrm{2y\:=\:55\:-\:7\:=\:48}$$

$$\mathrm{y\:=\:24}$$

Substitute the y value in equation (3)

$$\mathrm{x\:=\:7\:+\:24\:=\:31}$$

Therefore $\mathrm{31\:+\:24\:=\:55\:and\:31\:-\:24\:=\:7}$

Here $\mathrm{y\:=\:24}$ is the even number. Because the remainder value becomes zero when 24 is divided by 2.

2)If the multiplication of two numbers is 27 which is an odd number and the subtraction of those numbers is an even number. Find the numbers.

Ans

From the given instruction,

$\mathrm{x\times\:y\:=\:27\:,\:(x\:>\:y)}$

27 is a composite number. Because it can be divided by other than 1 and 27. So the values of 𝑥 𝑎𝑛𝑑 𝑦 will be 9 and 3. Because the divisors of 27 are 1,3,9 and 27. From the given instruction 𝑥 − 𝑦 will be an even number.

Therefore, from the subtraction property of even numbers $\mathrm{Odd\:-\:Odd\:=\:Even}$. 9 and 3 are odd numbers.

$\mathrm{9\:-\:3\:=\:6}$ . 6 is an even number.

Conclusion

Mathematics is a branch of science that arose due to the needs of man for calculation in business, knowing the relationship between numbers, measuring the land, in predicting cosmic events. The notation used to represent numbers is called numerical.

The development of the concept of numbers is the origin of mathematics. Even numbers are numbers that do not leave a remainder when divided by two. Multiples of two are even numbers. The most familiar numbers in practice, the numbers used for counting, are the natural numbers. Adding positive and negative numbers to natural numbers forms the integer.

FAQs

1. What are complex numbers?

In mathematics, a complex number is a combination of a real number and an imaginary number. If a and b denote two real numbers, then the complex number c is denoted as follows

$$\mathrm{c\:=\:a\:+\:bi}$$

2. Explain the numeral system?

The Numeral system (or system of numeration) refers to the writing system used to represent numbers. That is, it is a mathematical notation that represents a particular set of numbers as an integral, either numerically or by other symbols.

3. What are composite numbers?

A composite number is a positive integer that has at least one positive divisor (factor) other than the same number and one. That is, a composite number can be said to be a positive integer with no prime number greater than one.

4. What are odd numbers?

An odd number is a whole number whose last digits are an odd number, such as 1, 3, 5, 7, 9. The sum and difference of two odd numbers are equal, but the product is odd.

5. Define Set ?

A set is an aggregate or sum of different items. Although this may seem like a very simple concept, it is one of the most profound fundamental concepts in mathematics. The components in the set are called elements.