Real Numbers and Their Decimal Expansions

Rational and irrational numbers, as well as approximations of irrational numbers by rational numbers, can all be conveniently represented using the decimal expansion of a real number. Also, we are able to explore a real number's properties in greater detail because of the connections between the method of obtaining the decimal expansion of a real number and dynamical systems and probability theory. In this presentation, a brief introduction is given to the real number representation using decimal expansion and illustrates some key characteristics.

Real Numbers Expanded in Decimal: Definition

In order to write real numbers in decimal form, which has values that are numerically equivalent to real numbers, one must first simplify the real numbers even more. Using the long division technique, we can write the decimal expansion of real numbers.

By dividing the denominator by the numerator of the real number using the long division method, you can determine the decimal value of the given real numbers by looking at the quotient that results.

Real Numbers

Real numbers, on the other hand, are those that can be represented on a number line. Any numbers that may be discovered in the real world are referred to as real numbers, to put it simply. Real numbers in mathematics are the collection of both rational and irrational numbers.

Rational numbers are those that can be expressed as $\dfrac{a}{b}$ , $b \ne 0$ where $a$ and $b$ are both integers. Irrational numbers are those that cannot be expressed as the symbol $\dfrac{a}{b}$. Real numbers that are not rational are referred to as irrational numbers.

Real Numbers Examples

$\sqrt {11} $ , $\sqrt 2 $ , $\pi $$ \ldots $ etc.

Real Numbers and Their Decimal Expansions

The long division method can be used to write the decimal expansion of real numbers. By dividing the numerator of the real number by the denominator of the real number, the long division method can determine the decimal value of the given real numbers.

There are two different types of decimal expansions that can be used to represent any real number in the form of $\dfrac{p}{q}$, where $q \ne 0$. It is they

• Terminating Decimals

• Non-terminating Decimals

• The result is known as a terminating decimal if we divide the numerator by the denominator using the long division method and obtain a zero as the remainder.

• Non-terminating decimals are used in the division even when the resultant remainder is not zero and keeps going.

Decimal Representation of Terminating Numbers

A number that is obtained in decimal form and has a limited number of terms after the decimal point is said to be terminating. The terminating decimal number is the rational number's decimal expansion with a predetermined number of digits following the decimal point.

Every rational integer has a decimal point at the end. The number of terms after the decimal point in the rational number's decimal expansion is finite.

Using the rational number $\dfrac{1}{4}$ as an example, the decimal expansion \[\dfrac{1}{4} = 0.25\] is a terminating decimal since there are only two digits after the decimal point.

Decimal Representation of Non-terminating Numbers

Non-terminating decimal numbers are rational numbers that have an infinite or non-finite number of digits after the decimal point in the decimal expansion.

Non-terminating Decimals Come In Two Different Forms.

• Non-terminating, Recurring (Repeating) Decimals

• Non-terminating, Non-recurring (Non-repeating) Decimals

Non-terminating recurring decimals are ones in which one or more numbers, or patterns of numbers, continue after the decimal point. Such numbers are also known as rational numbers.

Example: The real number \[\dfrac{4}{9}\] decimal expansion is \[0.444444 \ldots \]

After the decimal point, the number $4$ is repeated. Such numbers are the non-terminating numbers represented in decimal expansion.

Decimal Expansion of Irrational Numbers

Irrational numbers are those that cannot be expressed in the form of $\dfrac{a}{b}$. Irrational numbers are real numbers that are not rational. Irrational numbers are any decimal representations of numbers that don't terminate or repeat.

For instance, the numbers $\sqrt 2 $ , $\sqrt 3 $ are irrational.

The number of terms or digits after the decimal point is limitless for irrational quantities. In the decimal expansion of the irrational numbers, one or more numbers or patterns of numbers do not repeat after the decimal point.

As a result, the expansion of irrational numbers into decimals is non-terminating and non-recurring.

An irrational number, Pi $(\pi ) = \dfrac{{22}}{7}$ , has a decimal expansion of \[3.145 \ldots \] which is non-terminating and non-recurring.

Theorems of Real Numbers Decimal Expansion

By using prime factorisation, we may determine the decimal expansion of any real number by determining the factors of the denominator in the form of $\dfrac{p}{q}$.

Theorem 1: If any real number is stated in the form $\dfrac{p}{q}$ and the prime factorisation of the denominator $q$ is expressed in the form ${2^m} \times {5^n}$ where $m$ and $n$ are non-negative integers, then we can claim that the decimal expansion of the real number is terminating.

as in \[\dfrac{3}{4} = 0.75\] .

Theorem 2: If any real number's decimal expansion in the form of $\dfrac{p}{q}$ is terminating by definition, then the denominator $q$ is stated in the form of ${2^m} \times {5^n}$, where $m$ and $n$ are non-negative integers.

An illustration is $\dfrac{3}{{40}} = \dfrac{3}{{{2^3} \times 5}}$ .

Theorem 3: If any real number can be represented as $\dfrac{p}{q}$ and the prime factorisation of the denominator $q$ cannot be expressed as ${2^m} \times {5^n}$, where $m$ and $n$ are non-negative integers, then we can conclude that the decimal expansion of the real number is non-terminating.

An example is $\dfrac{1}{6} = 0.1666 \ldots = 0.1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{6} $ .

Decimal Expansion Example

Q1. Demonstrate that the \[\dfrac{{14}}{{11}}\] decimal expansion is non-terminating and repeating.

Solution. \[\dfrac{{14}}{{11}}\] given as a number.

\[\dfrac{{14}}{{11}}\] is expanded to \[1.2727272727 \ldots \] decimals.

In this case, it is non-terminating and the number $27$ after the decimal point is repeating.

We can say that the provided number \[\dfrac{{14}}{{11}}\] is a rational number because it is non-terminating and repeating.

Thus, proved.

Q2. The rectangle has a length of $2.3$ feet and a width of $1.8$ feet. Determine if the rectangle's area is a terminating or non-terminating decimal.

Solution. Provided that the rectangle's length is $2.3$ feet.

The rectangle has a $1.8$ foot width.

We are aware that length $ \times $ breadth determines the rectangle's area.

Area \[ = 2.3 \times 1.8 = 4.14\] sq. feet

After the decimal point, there are definite digits.

As a result, the rectangle's area is obtained in terminating decimals.

Practice Questions

1. Express the real number $\dfrac{1}{12}$ in the form of decimal expansion.

Ans: $0.076923$

2. What would be the value of \[1.999\] if it is in the form $\dfrac{p}{q}$ , given \[q = 0\] and $p$ and $q$ both are integers?

Ans: $2$

Summary

We have explored real numbers in this article, which is a collection of rational and irrational numbers. We've talked about what real number decimal expansion means. This article also provides rational and irrational numbers' decimal expansions. We have looked at how recurring decimals and non-recurring (non-repeating) decimals expand to decimals. We have examined the real number decimal expansion theorems and worked through instances, which makes it easier for us to comprehend the topic.