What is meant by real numbers?

The thing about math is you have to learn terms in order to understand what numbers are, what kinds of numbers are out there, and the characteristics of each type. Numbers are just mathematical symbols that are used to count and measure. But not all numbers are created equal.

For instance, take the concept of "real numbers." If numbers can be real, are there also fake numbers? Well, yes — at least there are real numbers and imaginary numbers. But what does that mean?

Real Numbers Are All the Numbers

Real numbers are basically all the numbers you could think of if somebody told you to think of a number. Real numbers are based on the concept on the number line: the positive numbers sitting to the right of zero, and the negative numbers sitting to the left of zero. Any number that can be plotted on this number line is a real number. The numbers 27, -198.3, 0, 32/9 and 5 billion are all real numbers. Strangely enough, numbers such as √2 (the square root of 2, the value of which is 1.14142...) and π (3.1415...) can be plotted on a number line as well, even though they are nonterminating decimal numbers. So, even though the number after the decimal never ends, they can still be plotted on the number line.

Real numbers can also be described as all the numbers that are either rational or irrational. Rational numbers are numbers that can be written as a fraction, which includes whole numbers, all of which can be written as a fraction: 3/8, 5/1, 9/10, etc. Decimals can be rational as well — they're just numbers that have either terminating or repeating decimals. So, 8.372 is a terminating decimal and 5.2222222... is a repeating decimal. These are rational numbers, which are also real numbers. Irrational numbers are also real numbers: those are decimals that are nonterminating like π and √2.

In contrast, an imaginary number is the value of the square root of a negative number. You may remember this special little math rule, but there is no number that, when squared, will produce a negative number. But that doesn't stop mathematicians from doing it, as long as they admit the result is imaginary. Infinity is also an imaginary number.

Now That's Interesting

Real numbers were just "numbers" until the 1500s, when the Italian polymath Girolamo Cardano invented imaginary numbers in order to solve polynomial equations.

Real Numbers: The number system, often known as the numeral system, is a method of expressing numbers. There are two categories of the number system, i.e., real and imaginary numbers. Real numbers are the sum of rational and irrational numbers. All arithmetic operations may be done on these numbers in general. Furthermore, they can also be represented on a number line.

In comparison, imaginary numbers are unreal numbers that cannot be stated on a number line and are typically employed to represent complex numbers. Examples of real numbers are 23, -12, 6.99, 5/2, and so on. In this article, students will learn about the real numbers definition, their properties, chart, sets, and solved examples. Read on to find out more.

What are Real Numbers?

A combination of rational numbers and irrational numbers are known as real numbers. Real numbers can be both positive and negative, which is denoted as \(‘R’.\) natural numbers, fractions, and decimals all come under this category.

Types of Real Numbers

Different types along with real numbers examples are as given below:

1. Natural Numbers \(\left( N \right)\): Natural numbers are the counting numbers from \(1\). For example, they are \(1,2,3,4,5,…\) Natural numbers are a subset of real numbers. It is denoted by \(N\).
2. Whole Numbers \(\left( W \right)\) : Numbers \(0,1,2,3 \ldots \) are called whole numbers. These are natural numbers, including \(0\). Whole numbers are also a subset of real numbers. It is denoted by \(W\).
3. Integers \(\left( Z \right)\): The numbers …, \({\rm{ – 3, – 2, – 1,0,1,2,3,4,5,}}\) … are called integers. Integers are also a subset of real numbers. It is denoted by \(Z\).
4. Rational Numbers \(\left( Q \right)\) : A rational number is defined as a number that can be expressed in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime integers and \(q \ne 0.\). Rational numbers are also a subset of real numbers. It is denoted by \(Q\). Examples: \( – \frac{2}{3},0,5,\frac{3}{{10}}\), …. etc.
5. Irrational Numbers \(\left( P \right)\): An irrational number is defined as the number that cannot be expressed in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime integers and \(q \ne 0.\) Irrational numbers are also a subset of real numbers. It is denoted by \(P\). Examples: \(\sqrt 2 ,\sqrt 5 ,\pi ,e,\), …. etc.

What Is the Set of Real Numbers?

Now that you have understood the real numbers definition and examples, let’s see what constitute real numbers. The set of real numbers include natural numbers, whole numbers, integers, irrational numbers, and rational numbers, as defined and explained above.

What are the Properties of Real Numbers?

There are four main properties of real numbers. They are \(\left( {\rm{i}} \right)\) Commutative Property for addition and multiplication, \(\left( {\rm{ii}} \right)\) Associative Property for addition and multiplication, \(\left( {\rm{iii}} \right)\) Distributive Property of multiplication over addition and \(\left( {\rm{iv}} \right)\) Identity Property.

Commutative Property for Addition and Multiplication

The word commutative comes from “commute” or “move around”. Commutative properties mean that if the numbers we operate are changed or swapped from their position, the answer remains the same. This property is applicable only in the case of addition and multiplication but not for subtraction or division. If \(m\) and \(n\) are two real numbers, then,

For addition: \(m + n = n + m\), examples: \({\rm{6 + 3 = 9 = 3 + 6}}\) or \({\rm{2 + 8 = 10 = 8 + 2}}\)

For multiplication: \(m \times n = n \times m\), example: \(2 \times 4 = 8 = 4 \times 2\) or \(5 \times 10 = 50 = 10 \times 5\)

Associative Property for Addition and Multiplication

Associative Property states that you can add or multiply the numbers regardless of how they are grouped. If \(m,n\) and \(r\) are three real numbers, then,

For Addition: \(m + (n + r) = (m + n) + r\), example: \({\rm{3 + (4 + 5) = (3 + 4) + 5}}\)

For Multiplication: \((m \times n) \times r = m \times (n \times r)\), example: \((6 \times 2) \times 3 = 6 \times (2 \times 3)\)

If \(m,n\) and \(r\) are three real numbers, then the multiplication of real numbers is distributive over addition, and we can write \(m(n + r) = mn + mr\) and \((m + n)r = mr + nr\).
An example of distributive property is \(4(3 + 2) = 20 = 4 \times 3 + 4 \times 2.\)

Identity Property

Additive Identity

When any real number is added with \(0\), the answer is the same real number. If \(m\) is any real number, then \(m + 0 = m\). (\(0\) is the additive identity for real numbers).

Multiplicative Identity

When the number \(1\) is multiplied by any real number, the answer is the same real number. If \(m\) is any real number, then \(m \times 1 = 1 \times m = m\). \(1\) is the multiplicative identity for real numbers

What Is a Real Numbers Chart?

The chart is shown below, consisting of a set of real numbers that includes all the parts of real numbers:

Divisibility

A non-zero real number \(a\) is said to divide a real number \(b\) if there exists an integer \(c\) such that \(b = ac\).

The real numbers \(b\) is called the dividend, \(a\) is known as the divisor, and \(c\) is known as the quotient.

For example, \(3\) divides \(36\) because there is an integer \(12\) such that \(36 = 3 \times 12\). However, \(3\) does not divide \(35\) because there does not exist an integer \(c\)  such that \(35 = 3 \times c\). In other words, \(35 = 3 \times c\) is not true for any integer \(c\).

If a non-zero real number \(b\) divides another real number \(a\), then we write \(\frac{a}{b}\). We read it as \(b\)  divides \(a\). When \(\frac{a}{b}\)  is an integer, we say that \(a\) is divisible by \(b\)  or \(b\)  is a factor of \(a\) or \(a\) is a multiple of \(b\), or \(b\) is a divisor of \(a\).

We observe that:
i. \( – 4\) divides \(20\) because there exists an integer \( – 5\) such that \(20 =  – 4 \times ( – 5)\)
ii. \(4\) divides \( – 20\) because there exists an integer \( – 5\) such that \( – 20 = 4 \times ( – 5)\)

Euclid’s Division Lemma

Euclid is a Greek Mathematician who has made a lot of contributions to number theory. Among these, Euclid’s Lemma is the most important one. A Lemma is a proven statement that we use to prove other statements. This Lemma is nothing but a restatement of the long division process.

Theorem of Euclid’s Division Lemma

Let \(a\) and \(b\) be any two positive integers. Then there exist unique integers \(q\) and \(r\) such that

\(a = bq + r,0 \le r < b\)

If \(b\)  divides, \(a\)  then \(r = 0\). Otherwise, \(r\) satisfies the inequality \(0 < r < b\).

Euclid’s Division Algorithm

Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Now, recall that the HCF of two positive integers \(a\)  and \(b\)  is the largest positive integer d that divides both \(a\) and \(b\).

Fundamental Theorem of Arithmetic

We know that any natural number can be written as a product of its prime factors. For instance, \(2 = 2,4 = 2 \times 2,253 = 11 \times 23\), and so on. Can any natural number be obtained by multiplying prime numbers?

Take any collection of prime numbers, say \(2,3,7,11\)and \(23\). If we multiply some or all these numbers, allowing them to repeat as many times as we wish, we can produce an extensive collection of positive integers. (In fact, infinitely many) like as follows:

\(7 \times 11 \times 23 = 1771\)

\(2 \times 3 \times 7 \times 11 \times 23 = 10626\)

\(3 \times 7 \times 11 \times 23 = 5313\)

\({2^2} \times 3 \times 7 \times 11 \times 23 = 21252\)

\({2^3} \times 3 \times {7^3} = 8232\) and so on.

There are infinitely many prime numbers. If you combine all these primes in all possible ways, then you get an infinite collection of numbers, all the primes and all possible products of primes. Can you produce all the composite numbers this way? Do you think that there may be a composite number that is not the product of powers of prime? Let us factorise positive integers.

Theorem

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

The Fundamental Theorem of Arithmetic has many applications, both within mathematics and in other fields.

Solved Examples – Real Numbers

Q.1. Show that any positive integer is of form \(3q\) or \(3q + 1\) or \(3q + 2\) for some integer \(q\).
Sol: Let \(a\) be any positive integer and \(b = 3.\) Applying division lemma with \(a\) and \(b = 3\), we have
\(a = 3{\rm{ }}q + r\) where \(0 \le r < 3\) and \(q\) is some integer.
\( \Rightarrow a = 3q + 0\) or \(a = 3{\rm{ }}q + 1\) or \(a = 3{\rm{ }}q + 2\)
\( \Rightarrow a = 3q\) or \(a = 3{\rm{ }}q + 1\) or \(a = 3{\rm{ }}q + 2\) for some integer \(q\)

Q.2. Show that any positive odd integer is of form \(4q + 1\) or \({\rm{4}}q + 3\) where \(q\) is some integer
Sol: Let \(a\) be any odd positive integer and \(b = 4\). By division Lemma there exists integers \(q\) and \(r\) such that
\(a = 4{\rm{ }}q + r\), where \(0 \le r < 4\)
\( \Rightarrow a = 4q\) or, \(a = 4{\rm{ }}q + 1\) or, \(a = 4{\rm{ }}q + 2\)or, \(a = 4{\rm{ }}q + 3\) \([0 \le r < 4 \Rightarrow r = 0,1,2,3]\)
\( \Rightarrow a = 4q + 1\) or, \(a = 4{\rm{ }}q + 3\)
\(\therefore a\) is an odd integer . \(\therefore a \ne 4q,a \ne 4q + 2\)
Hence, any odd integer is of form \(4{\rm{ }}q + 1\) or \(4q + 3\).

Q.3. Prove that \(x\) and \(y\) are odd positive integers, then \({x^2} + {y^2}\) is even but not divisible by \(4\).
Sol: We know that any odd positive integer is of the form \(2q + 1\) for some integer \(q\).
So, let \(x = 2m + 1\) and \(y = 2n + 1\) for some integers \(m\) and \(n\).
\(\therefore \quad {x^2} + {y^2} = {(2m + 1)^2} + {(2n + 1)^2}\)
\( \Rightarrow \quad {x^2} + {y^2} = 4\left( {{m^2} + {n^2}} \right) + 4(m + n) + 2\)
\( \Rightarrow \quad {x^2} + {y^2} = 4\left( {{m^2} + {n^2}} \right) + (m + n) + 2\)
\( \Rightarrow \quad {x^2} + {y^2} = 4q + 2,\) where \(q = \left( {{m^2} + {n^2}} \right) + (m + n)\)
\( \Rightarrow \quad {x^2} + {y^2}\) is even and leaves remainder \(2\) when divided by \(4\)
\( \Rightarrow \quad {x^2} + {y^2}\) is even but not divisible by \(4\)

Q.4. Find the HCF of  \(96\) and \(404\) by the prime factorisation method. Hence, find their LCM.
Solution: The prime factorisation of \(96\)  and \(404\)  gives:
\(96 = {2^5} \times 3,404 = {2^2} \times 101\)
Therefore, the \({\rm{HCF}}\) of these two integers is \({2^2} = 4\)
Also, \({\rm{LCM}}\) \((96,404) = \frac{{96 \times 404}}{{{\rm{HCF}}(96,404)}} = \frac{{96 \times 404}}{4} = 9696\)

Q.5. Find the \({\rm{HCF}}\) and \({\rm{LCM}}\) of \(6,72\) and \(120\), using the prime factorisation method.
Solution: We have: \(6 = 2 \times 3,72 = {2^3} \times {3^2},120 = {2^3} \times 3 \times 5\)
Here, \({2^1}\) and \({3^1}\) are the smallest powers of the common factors \(2\) and \(3\), respectively.
So, \({\rm{HCF}}\) \((6,72,120) = {2^1} \times {3^1} = 2 \times 3 = 6\)
\({2^3},{3^2}\) and \({5^1}\) are the greatest powers for the prime factors \(2,3\) and \(5\) respectively involved in the three numbers.
So, \({\mathop{\rm LCM}\nolimits} (6,72,120) = {2^3} \times {3^2} \times {5^1} = 360\)
Remark
Notice \(6 \times 72 \times 120 \ne {\rm{HCF}}(6,72,120) \times {\rm{LCM}}(6,72,120)\) So, the product of three numbers is not equal to the product of their \({\rm{HCF}}\) and \({\rm{LCM}}{\rm{.}}\)

Summary

In the given article, the topics covered are the definition of real numbers, types of real numbers, set of real numbers, real number chart, etc. Then, we discussed Divisibility, Euclid’s division Lemma along with theorem and proof, then explained about Euclid’s division algorithm, and the fundamental theorem of arithmetic along with the theorem. Later the solved examples are given, followed by frequently asked questions. Real numbers are the backbone for understanding number systems and aid in mathematical calculations in all levels of mathematics.

How do you define real numbers?

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