Properties of Real Number: Real numbers are any numbers one can think of. In other words, real numbers are all the numbers that are rational such as fractions, or irrational such as decimals. The term properties in this context defines the characteristics of the real numbers below the arithmetic operations of addition and multiplication that are gained even without proof. You will see the various properties of the real numbers \({\rm{(R)}}\). Understanding the properties of real numbers will help you simplify the numerical and algebraic expressions, solve the equations, etc.

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CLEAR YOUR CONCEPTUAL DOUBTS ON REAL NUMBERS

What are Real Numbers?

Definition: The combination of rational numbers and irrational numbers is known as real numbers. Real numbers can both positive or negative and are denoted as \({\rm{‘R}}\). Natural numbers, fractions, and decimals all come under real numbers.

What are Types of Real Numbers?

Several types of real numbers are given below:

Natural Numbers	Natural numbers are the real numbers that do not have any decimal and are greater than zero.
Whole Numbers	Whole numbers are positive real numbers, and they do not have decimals and zero. Natural numbers are also considered whole numbers.
Integers	Integers have no decimals, and they include both positive and negative numbers. Even whole numbers are also integers.
Rational Numbers	Rational numbers are the real numbers that can be written in fractions using integers. Integers are also known as rational numbers.
Irrational Numbers	Irrational numbers are real numbers that cannot be written in fraction using integers. Transcendental numbers are also considered irrational numbers.
Transcendental Numbers	These numbers cannot be obtained by solving an equation with the integer components.

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What are the Properties of Real Numbers?

There are four main properties of the real number: commutative property, associative property, distributive property, and identify the property.

The three real numbers are \(‘m’,’n’\) and \(‘r’\) The properties are described using the three real numbers \(m,\,n\) and \(r\)

Addition Properties of Real Numbers

What are the five properties of real numbers?

The five properties addition properties of real numbers are given below:

1. Closure Property
2. Commutative Property
3. Associative Property
4. Additive Identity Property
5. Additive Inverse Property

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Closure Property

The formula is \(a + b\) is the real numbers. The sum of two real numbers is also a real number.

Example: \(9 + 7 = 16\) Here \(9\) and \(7\) are real numbers, and their sum \(16\) is also a real number.

Commutative Property

The commutative property states that numbers can be added in any order, and the result remains the same.

Example:

1. There are two groups, the first group has one \((4)\) chocolates, and the other group has \((6)\)chocolates. When you add, you will get a total of \((10)\) chocolates.
2. Here, what if we changed the distributions. The first group now has \((6)\) chocolate and the second group has \((4)\) chocolate. How many would be the total?
3. The total would be still the same \((10)\)
4. In the commutative property of addition, no matter which group has \((6)\) chocolates and the \((4)\) chocolates, the total would be still the same, i.e.,\((10)\) chocolates.

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Associative Property

The associative property states that regardless of how numbers or objects are grouped, the total sum would still be the same.

Example: We will add the number of teddies Pari, Preet and Sana using the expression \((A + B) + C\)

1. Pari has \(4\) teddies while Preet has \(3\) teddies. Hence there are a total of \(7\) teddies. Let us suppose Sana has \(2\) teddies,  will add the number of teddies to the total number of teddies; \((4 + 3) + 2 = 9\)
2. Now, let us try applying the associative property.
3. Following \((A + B) + C\), we will add the number of teddies Preet and Sana have; we \(5\) teddies.
4. Then, add the number of teddies Pari has, which is \(4\) then we get \(9\).
5. The total number of teddies are \(9\).
6. Regardless of whether we added the number of teddies Pari and Preet have, then add Sana’s number of teddies or add Preet and Sana’s teddies first and then add Pari’s teddies. Again, this is due to the associative property.

Additive Identity Property

Another property states that the sum of any number and zero is the number itself.

Example:

1. Rahul has \(4\) balloons while his friend does not have any balloon; thus, \(0\).
2. Adding \(0\) to any number does not change the value of the number.

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Additive Inverse Property

The additive inverse of a number is a number that results in zero when added to the original number \((0)\).

Examples: We see that \(7 + ( – 7) = 0\) So, you can consider that the number \( – 7\) is the additive \(7\).

Also, \( – 15 + (15) = 0\) So, \(15\) is the additive inverse of \(-15\).

Multiplication Properties of Real Numbers

The properties of multiplication of real numbers are given below:

Closure Property

The formula is \(a\, \times \,b\) is the real numbers. When you multiply any two real numbers, the product which you get is also a real number.

Example: \(8\, \times \,7 = 56\) in which \(56\) is the product and a real number similar to the numbers \(8\) and \(7\).

Commutative Property

The formula \(a \times b = b \times a\) holds for the commutative property.

When you multiply any two real numbers in any order, the product you get will always be the same.

Example: \(8 \times 9 = 9 \times 8 = 72;\) you have interchanged the numbers \(8\) and \(7\) but the product is the same.

Associative Property

In this property, the formula is \((a \times b) \times c = a \times (b \times c)\)

That means when you are multiplying any three real numbers, the product will be the same no matter how you group them.

Example: \((2 \times 3) \times 4 = 2 \times (3 \times 4) = 24\) you can see the product is the same even when you have changed the grouping of the numbers.

Multiplication Identity Property

In this property, the formula is \(a \times 1 = a\)

Whenever you multiply any real number with the number one, you will get the same number as the product.

Example: \(68 \times 1 = 68\) or you can interchange the numbers \(1 \times 68 = 68\) you can see that you have got the same number as the product.

Multiplication Inverse Property

In this property, the formula is \(a \times \frac{1}{a} = 1\) but \(a \ne 0\)

Whenever you multiply any non zero real number by its inverse or the reciprocal, you will get the product as one always.

Example: \(3 \times \frac{1}{3} = 1\)

Distributive Property of Multiplication

This property is entirely different from the Commutative and the Associative property. Here, the sum of two numbers multiplied by the third number equals the sum when each number is multiplied by the third number.

\(A \times (B + C) = A \times B + A \times C\)

In this case, \(A\) is the monomial factor and \((B + C)\) is the binomial factor.

Example:

Let’s take \(A = 1,B = 4\) and \(C = 2\)

\({\rm{L}}{\rm{.H}}{\rm{.S}} = A \times (B + C) = 1 \times (4 + 2)\)
\( = 1 \times 6\)
\( = 6\)
\({\rm{R}}{\rm{.H}}{\rm{.S}} = A \times B + A \times C = 1 \times 4 + 1 \times 2\)
\( = 4 + 2\)
\( = 6\)
\({\rm{L}}{\rm{.H}}{\rm{.S}}\,{\rm{ = R}}{\rm{.H}}{\rm{.S}}\)
\(6 = 6\)

In the given above example, you can see, even when distributed \(A\) (monomial factor) to each value of the binomial factor \(B\) and \(C\) the value remains the same on both sides. Thus, the distributive property is significant as it combines both the addition operation and the multiplication operation.

Combined Property of Multiplication Together with Addition

Distributive Property of Multiplication over Addition

If \(a,b\) and \(c\) represent the real numbers:

Then the formula for this property will be \(a(b\, + c) = \,ab + ac\) or \(\left({a + b} \right)c = ac + bc\)

Here, you can see the operation of the multiplication distributes over the operation of the addition.

Example: \(4\left({5 + 8} \right) = 4 \times 5 + 4 \times 8\) or \(\left({5 + 8} \right)4 = 5 \times 4 + 8 \times 4\)

Solved Examples – Properties of Real Number

Q.1. Multiply \(475\) and by \(475\) by using the distributivity property.
Ans: We have, \(475 \times 64\)
\( = (400 + 70 + 5) \times 64\)
\( = 400 \times 64 + 70 \times 64 + 5 \times 64\) [Using distributivity]
\( = 25600 \times 4480 + 320 = 30400\)

Q.2. Find the product \(4 \times 4957 \times 25\)
Ans: We observe that,
\(4 \times 25 = 100\)
So, we can arrange the numbers to find the desired product
\(4 \times 4957 \times 25 = (4 \times 25) \times 4957 = 100 \times 4957 = 495700\)
Hence, the required answer is \(495700\)

Q.3. Evaluate \(9 – 10 + ( – 5) + 6\)
Ans: Given, \(9 – 10 + ( – 5) + 6\)
First, we will open the brackets.
\(9 – 10 – 5 + 6\)
Now, we will add the positive and negative integers separately.
\(9 + 6 – 10 – 5\)
\( = 15 – 15\)
\( = 0\)

Q.4 Add the given numbers \(43 + 56 + 23\) using the associative property.
Ans: Given \(43 + 56 + 23\)
Now add the numbers using the associative property \((a + b) + c = a + (b + c)\)
So, using the property, you write \((43 + 56) + 23 = 43 + (56 + 23)\)
Here, \(99 + 23 = 43 + 79\)
\(122 = 122\)

Q.5. Multiply the given numbers \(12 \times 6 \times 20\)
Ans: Given, \(12 \times 6 \times 20\)
Now add the numbers using the associative property \((a \times b) \times c = a \times (b \times
c)\)
So, using the property, you write \((12 \times 6) \times 20 = 12 \times (6 \times 20)\)
Here, \(72 \times 20 = 12 \times 120\)
\(1440 = 1440\)

Summary

In the given article, we have discussed real numbers and their types. The properties of real numbers, such as Closure Property, Commutative Property, Associative Property, Associative Identity Property, Additive Inverse Property, are discussed along with their respective examples.  The multiplication properties for real numbers and examples helped us understand the concept clearly.

FAQs on Properties of Real Number

Q.1. What are the five properties of real numbers?
Ans: The five properties of real numbers are:
1.Closure Property
2. Commutative Property
3. Associative Property
4. Additive Identity Property
5. Additive Inverse Property

Q.2. Why are the properties of real numbers important when factoring?
Ans: The properties of the real number are essential when factoring because you will not be able to solve the whole factoring solution if there are no real numbers.

Q.3. Why do we need properties of numbers?
Ans: We need properties of the numbers as they are essential because you use them consistently in pre-calculus. The name does not often use the properties in pre-calculus, but you are supposed to know when you need to use them.

Q.4. What is the formula of commutative property?
Ans: In this property of commutative, the formula is \(a \times b = b \times a\)
When you multiply any two real numbers in any order, the product you get will always be the same.
Example: \(8 \times 9 = 9 \times 8 = 72\) you have interchanged the numbers \(8\) and \(9\) but the product is the same.

Q.5. Which property defines that by multiplying any number by \(1\) gives the same number?
Ans: It is Multiplicative Identity Property.
In this property, the formula is \(a \times 1 = a\)
Whenever you multiply any real number with the number one, you will get the same number as the product.
Example: \(68 \times 1 = 68\) or you can interchange the numbers \(1 \times 68 = 68\) you can see that you have got the same number as the product.

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