Decimal Expansions and Rationality

Terminating and Non-Terminating Decimal Expansions

The Decimal Representation of Numbers

Every real number can be uniquely represented as a decimal expansion. This is the representation of the number in base 10, using the digits 0 through 9 and a decimal point. The decimal point separates the integer part of the number (to the left) from the fractional part (to the right).

The position of each digit relative to the decimal point determines its place value, which is a power of 10. Digits to the left of the decimal represent non-negative powers of 10 ($10^0, 10^1, 10^2, ...$), while digits to the right represent negative powers of 10 ($10^{-1}, 10^{-2}, 10^{-3}, ...$).

For a rational number, which is defined as any number that can be written as a fraction $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$), its decimal expansion is obtained by performing the long division of the numerator $p$ by the denominator $q$. The characteristics of this decimal expansion are key to understanding the nature of rational numbers and distinguishing them from irrational numbers.

During the process of long division $p \div q$, we generate digits after the decimal point based on the remainders at each step. Since we are dividing by $q$, the only possible remainders at any step are integers from $0$ up to $|q|-1$. There is a finite number of these possible remainders.

Based on the behavior of these remainders, the decimal expansion will either end (terminate) or continue infinitely (non-terminate).

Terminating Decimal Expansions

A decimal expansion is called terminating if the long division process of the numerator by the denominator eventually results in a remainder of $0$. When the remainder becomes 0, the division stops, and the decimal representation has a finite number of digits after the decimal point.

Examples:

• $\frac{1}{2}$: When dividing 1 by 2, we get 0.5 with a remainder of 0. Thus, $\frac{1}{2} = 0.5$.
• $\frac{3}{4}$: When dividing 3 by 4, we get 0.75 with a remainder of 0. Thus, $\frac{3}{4} = 0.75$.
• $\frac{7}{8}$: When dividing 7 by 8, we get 0.875 with a remainder of 0. Thus, $\frac{7}{8} = 0.875$.
• $\frac{13}{100}$: This is already a fraction with a power of 10 as the denominator, representing 13 hundredths. $\frac{13}{100} = 0.13$.

Terminating decimals are always rational numbers, as they can be written as a fraction with a denominator that is a power of 10 (e.g., $0.75 = \frac{75}{100}$).

The Condition for Terminating Decimal Expansions

A rational number, expressed as a fraction $\frac{p}{q}$ in its simplest form (meaning the numerator $p$ and the denominator $q$ have no common factors other than 1, i.e., $\text{GCD}(p, q) = 1$), has a terminating decimal expansion if and only if the prime factorization of its denominator $q$ contains only the prime numbers $2$ and/or $5$.

Explanation: The decimal number system is based on powers of $10$. A terminating decimal is fundamentally a fraction whose denominator can be expressed as a power of $10$. Since $10$ can be factored into its prime factors as $10 = 2 \times 5$, any positive integer power of $10$, say $10^k$, can be factored as $(2 \times 5)^k = 2^k \times 5^k$.

If the denominator $q$ of a fraction $\frac{p}{q}$ (in simplest form) has only prime factors of $2$ and/or $5$, it means $q$ can be written in the form $q = 2^m \times 5^n$ for some non-negative integers $m$ and $n$. We can make the denominator a power of 10 by multiplying the numerator and the denominator by an appropriate power of 2 or 5:

• If $m \ge n$, we need $m-n$ more factors of 5 in the denominator to match the number of factors of 2. We multiply by $\frac{5^{m-n}}{5^{m-n}}$: $$ \frac{p}{2^m 5^n} = \frac{p \times 5^{m-n}}{2^m 5^n \times 5^{m-n}} = \frac{p \times 5^{m-n}}{2^m 5^{m}} = \frac{p \times 5^{m-n}}{(2 \times 5)^m} = \frac{p \times 5^{m-n}}{10^m} $$
• If $n > m$, we need $n-m$ more factors of 2 in the denominator to match the number of factors of 5. We multiply by $\frac{2^{n-m}}{2^{n-m}}$: $$ \frac{p}{2^m 5^n} = \frac{p \times 2^{n-m}}{2^m 5^n \times 2^{n-m}} = \frac{p \times 2^{n-m}}{2^n 5^{n}} = \frac{p \times 2^{n-m}}{(2 \times 5)^n} = \frac{p \times 2^{n-m}}{10^n} $$

In both cases, the original fraction $\frac{p}{q}$ is equivalent to a fraction whose denominator is a power of $10$. Any fraction with a denominator that is a power of $10$ corresponds to a terminating decimal (e.g., $\frac{N}{1000} = 0.00N$).

Conversely, if the denominator $q$ of a fraction $\frac{p}{q}$ (in simplest form) contains any prime factor other than $2$ or $5$ (such as $3, 7, 11, 13$, etc.), it is impossible to multiply the denominator by any integer to make it a power of $10$, because a power of $10$ only has prime factors 2 and 5. Therefore, the decimal expansion will not terminate.

Example: $\frac{1}{3}$. Denominator is $3$. Prime factor is $3$. $\frac{1}{3} = 0.333...$ (non-terminating).
Example: $\frac{5}{6}$. In simplest form. Denominator is $6 = 2 \times 3$. Prime factors include $3$. $\frac{5}{6} = 0.8333...$ (non-terminating).
Example: $\frac{3}{7}$. In simplest form. Denominator is $7$. Prime factor is $7$. $\frac{3}{7} = 0.428571428571...$ (non-terminating).

Non-Terminating Decimal Expansions

A decimal expansion is called non-terminating if the long division process never produces a remainder of $0$. The digits after the decimal point continue indefinitely.

This happens when, during the division of $p$ by $q$, a remainder repeats at some point. Since there are only a finite number of possible non-zero remainders ($1, 2, ..., |q|-1$), if the remainder 0 is never reached, a remainder must eventually repeat. When a remainder repeats, the sequence of subsequent digits in the quotient will also repeat.

Non-terminating decimals fall into two categories:

• Non-terminating Repeating (or Recurring) Decimals: These are non-terminating decimals where a block of one or more digits repeats infinitely in the decimal expansion. These decimals correspond to rational numbers whose denominators (in simplest form) contain prime factors other than 2 or 5.

  Examples: $0.333... = 0.\overline{3}$ (repeating block is '3'), $\frac{1}{3}$.

  $0.142857142857... = 0.\overline{142857}$ (repeating block is '142857'), $\frac{1}{7}$.

  $0.454545... = 0.\overline{45}$ (repeating block is '45'), $\frac{5}{11}$.

  $0.1666... = 0.1\overline{6}$ (repeating block is '6' after initial non-repeating '1'), $\frac{1}{6}$.

  These decimals can always be converted back into the $\frac{p}{q}$ form, confirming they are rational.

• Non-terminating Non-repeating Decimals: These are non-terminating decimals where no block of digits repeats periodically. The digits continue infinitely without any discernible cycle. These decimals correspond to numbers that cannot be expressed as a fraction $\frac{p}{q}$.

  Examples: The decimal expansions of irrational numbers such as $\sqrt{2} \approx 1.41421356...$, $\sqrt{3} \approx 1.73205080...$, $\pi \approx 3.14159265...$, $e \approx 2.718281828...$. Also, consciously constructed non-repeating patterns like $0.01001000100001...$.

  These decimals represent the set of irrational numbers.

Summary: Decimal Expansions and Rationality

The type of decimal expansion a number has provides a definitive test for whether it is rational or irrational:

• A number is rational if and only if its decimal expansion is either terminating or non-terminating and repeating.
• A number is irrational if and only if its decimal expansion is non-terminating and non-repeating.

This property is a cornerstone in understanding the distinction between rational and irrational numbers on the real number line.

Decimal Expansion of Rational vs. Irrational Numbers

The type of decimal expansion a real number has is a fundamental property that provides a perfect distinction between rational and irrational numbers. Every real number can be expressed as a decimal, and the nature of this decimal (whether it terminates or repeats) tells us whether the number is rational or not.

Decimal Expansion of Rational Numbers ($\mathbb{Q}$)

A number is a rational number if and only if its decimal expansion is either terminating or non-terminating and repeating (recurring).

• Terminating Decimal Expansions:

  As discussed earlier, these are decimals that end after a finite number of digits. They correspond to rational numbers $\frac{p}{q}$ (in simplest form) where the denominator $q$ has only prime factors of 2 and/or 5.

  Examples:

  • $0.25 = \frac{25}{100} = \frac{1}{4}$. Denominator 4 has prime factors $2^2$.
  • $1.5 = \frac{15}{10} = \frac{3}{2}$. Denominator 2 has prime factor 2.
  • $0.125 = \frac{125}{1000} = \frac{1}{8}$. Denominator 8 has prime factors $2^3$.
  • $4.0 = \frac{4}{1}$. Denominator 1 has prime factors $2^0 5^0$.

• Non-Terminating Repeating Decimal Expansions:

  These are decimals that continue infinitely with a repeating sequence (block) of digits. These correspond to rational numbers $\frac{p}{q}$ (in simplest form) where the denominator $q$ contains at least one prime factor other than 2 or 5.

  In the long division of $p$ by $q$, since the division does not terminate (remainder never becomes 0) and there are only a finite number of possible non-zero remainders (from 1 to $|q|-1$), a remainder must eventually repeat. Once a remainder repeats, the sequence of digits in the quotient from that point onwards will also repeat in the same order. The repeating sequence of digits is called the period of the decimal expansion.

  The maximum possible length of the repeating block for $\frac{p}{q}$ (in simplest form) is $|q|-1$. For example, $\frac{1}{7}$ has a repeating block of length $6 = 7-1$.

  Examples:

  • $0.333... = 0.\overline{3} = \frac{1}{3}$. Denominator 3, prime factor 3. Period '3'.
  • $0.142857142857... = 0.\overline{142857} = \frac{1}{7}$. Denominator 7, prime factor 7. Period '142857'.
  • $0.090909... = 0.\overline{09} = \frac{1}{11}$. Denominator 11, prime factor 11. Period '09'.
  • $2.1666... = 2.1\overline{6} = \frac{13}{6}$. Denominator 6 ($=2 \times 3$), prime factors 2 and 3. '1' is the non-repeating part, '6' is the repeating part.
  • $0.454545... = 0.\overline{45} = \frac{5}{11}$. Denominator 11, prime factor 11. Period '45'.

The key takeaway is that the decimal expansion of a rational number is predictable; it either stops or it repeats a pattern.

Decimal Expansion of Irrational Numbers ($\mathbb{I}$)

A number is an irrational number if and only if its decimal expansion is non-terminating and non-repeating.

The digits after the decimal point continue infinitely without settling into any repeating pattern or block of digits.

Examples:

• $\sqrt{2} = 1.4142135623730950488016887242097...$ (The digits continue indefinitely without repeating).
• $\sqrt{3} = 1.7320508106795680557...$
• $\pi = 3.14159265358979323846...$
• $e = 2.71828182845904523536...$
• Numbers specifically constructed to have non-repeating patterns: $$ 0.10110111011110... $$ $$ 0.1234567891011121314... $$ $$ 0.202002000200002... $$ (In these examples, the pattern is clear, but it's designed specifically so no block of digits repeats, making them non-repeating decimals).

The property of having a non-terminating and non-repeating decimal expansion is the defining characteristic of irrational numbers in terms of their decimal form. This is precisely why they cannot be written as a simple fraction $\frac{p}{q}$. The decimal expansion provides a way to uniquely pinpoint the location of each irrational number on the continuous real number line.

Summary of the Relationship:

• Decimal is Terminating OR Non-terminating Repeating $\iff$ Number is Rational.
• Decimal is Non-terminating AND Non-repeating $\iff$ Number is Irrational.

This definitive link between the type of decimal expansion and the nature of the number (rational or irrational) is a fundamental concept in understanding the structure of the real number system.

Expressing Recurring Decimals in p/q Form

As established in the previous sections, a number is rational if and only if its decimal expansion is either terminating or non-terminating and repeating (recurring). Terminating decimals are easily converted to fractions by using powers of 10 as the denominator (e.g., $0.75 = \frac{75}{100}$). Non-terminating repeating decimals, however, require a different method to be converted into the standard fractional form $\frac{p}{q}$. This conversion process uses basic algebra.

Algebraic Method for Converting Recurring Decimals to Fractions ($\frac{p}{q}$)

The method relies on manipulating equations derived from the decimal representation by multiplying by appropriate powers of 10 to align the repeating parts, which are then eliminated by subtraction.

Consider a non-terminating repeating decimal number.

Steps:

1. Assign a variable: Let the given recurring decimal number be represented by a variable, say $x$. Write out its decimal expansion showing enough digits to clearly identify the repeating pattern.
2. Isolate the repeating block (Equation 1): If there is a non-repeating part immediately following the decimal point, multiply the original equation ($x = \text{decimal}$) by a power of $10$ ($10^k$) such that the decimal point moves just to the right of the non-repeating part and immediately before the beginning of the repeating block. Here, $k$ is the number of digits in the non-repeating part after the decimal point. Let this equation be denoted as Equation (1). (If there is no non-repeating part after the decimal, i.e., $k=0$, then Equation (1) is simply the original equation, $x = \text{decimal}$).
3. Shift one repeating block (Equation 2): Count the number of digits in the repeating block. Let this number be $m$. Multiply Equation (1) (the equation where the decimal point is just before the repeating block) by a power of $10$ ($10^m$) such that the decimal point moves to the right by $m$ places. This effectively shifts one entire block of repeating digits to the left of the decimal point. Let this new equation be denoted as Equation (2).
4. Eliminate the repeating part: Subtract Equation (1) from Equation (2). Because of the way the decimal points were aligned and the repeating blocks were shifted, the infinite repeating parts of the decimals in both equations will be identical after the decimal point. Therefore, these repeating parts will cancel each other out upon subtraction. This leaves a simple linear equation involving $x$ and integers.
5. Solve for the variable: Solve the resulting linear equation for $x$. This will express $x$ as a fraction $\frac{\text{integer}}{\text{integer}}$.
6. Simplify the fraction: Simplify the resulting fraction to its lowest terms by dividing the numerator and the denominator by their Greatest Common Divisor (GCD). This gives the fraction in the desired $\frac{p}{q}$ form, where $\text{GCD}(p, q) = 1$.

Examples

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This algebraic method confirms that any number with a non-terminating repeating decimal expansion is indeed a rational number, as it can be written as a fraction of two integers.

Proofs of Irrationality

A number is classified as irrational if it cannot be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Proving that a number is irrational thus involves demonstrating that no such fractional representation exists. Because irrationality is defined by the absence of a property (being rational), proofs of irrationality typically employ indirect methods, most notably proof by contradiction.

Proof by Contradiction Method

Proof by contradiction (also known as reductio ad absurdum, meaning "reduction to absurdity") is a fundamental logical technique in mathematics. It is used to prove a statement is true by showing that assuming its negation leads to a logical inconsistency or a contradiction.

The steps involved in a proof by contradiction are:

1. Clearly state the statement you want to prove. Let's call this statement P. (e.g., P: "The number $\sqrt{2}$ is irrational").
2. Assume the negation of statement P is true. This is your starting assumption for the contradiction. (e.g., Assume not P: "Assume $\sqrt{2}$ is rational").
3. Using this assumption, along with established definitions, axioms, proven theorems, and logical rules, derive a sequence of logical consequences.
4. Continue the derivation until you arrive at a statement that is a contradiction. A contradiction is a statement that is logically impossible or that directly conflicts with known facts, previously proven theorems, or even the initial assumption made in step 2. (e.g., concluding that an integer must be both even and odd, or that a fraction in simplest form has a common factor greater than 1).
5. Since the logical steps followed from the assumption were valid, the only way a contradiction could arise is if the initial assumption (that not P is true) was false.
6. Therefore, the initial assumption (not P) is false, which implies that the original statement (P) must be true.

Proof that $\sqrt{2}$ is Irrational

The proof that $\sqrt{2}$ is irrational is a classic example of a proof by contradiction and a foundational result demonstrating the existence of irrational numbers. It was known to the ancient Greek mathematicians.

Proof:

Proof that $\sqrt{3}$ is Irrational (Generalization for $\sqrt{p}$, p prime)

The proof structure used for $\sqrt{2}$ can be easily adapted to prove the irrationality of the square root of any prime number. The key property is that for a prime number $p$, if $p$ divides the square of an integer ($a^2$), then $p$ must divide the integer itself ($a$).

Proof:

This proof structure holds for $\sqrt{p}$ where $p$ is any prime number. The property that if a prime $p$ divides $a^2$, it must divide $a$, is derived from the Fundamental Theorem of Arithmetic (Unique Prime Factorization).

For integers $n$ that are not perfect squares but are also not prime (e.g., $\sqrt{6}$, $\sqrt{10}$), the proof is similar but relies more directly on the unique prime factorization. If $\sqrt{n} = p/q$ in simplest form, then $nq^2 = p^2$. Consider the prime factorization of $n$. If $n$ is not a perfect square, at least one prime factor in its factorization must have an odd exponent. When you multiply $n$ by $q^2$, any prime factor in $q^2$ has an even exponent. Thus, in the prime factorization of $nq^2$, any prime factor from $n$ that had an odd exponent will still have an odd exponent. However, $p^2$ must have all prime factors with even exponents. This conflict in the exponents of prime factors between $nq^2$ and $p^2$ (which are equal numbers) leads to a contradiction of the uniqueness of prime factorization, proving $\sqrt{n}$ is irrational if $n$ is not a perfect square.

Proofs for Other Irrational Numbers ($\pi, e$, etc.)

Proving the irrationality of numbers like $\pi$ (Pi) and $e$ (Euler's number) is significantly more complex than for square roots of non-perfect squares. These proofs typically require concepts from calculus, analysis, and advanced number theory.

• Irrationality of $e$: One common proof uses the infinite series definition of $e$: $e = \sum\limits_{n=0}^{\infty} \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \ldots$. Assuming $e = p/q$ and showing that this leads to a contradiction (often by manipulating the series) proves that $e$ is irrational. For example, for large enough $q$, $q! \times (e - \sum_{n=0}^q \frac{1}{n!})$ must be an integer based on the assumption, but the remainder series can be shown to be strictly between 0 and 1, leading to a contradiction.
• Irrationality of $\pi$: Proofs for $\pi$ are generally more intricate. One notable proof by Johann Heinrich Lambert (1761) used continued fractions. Later proofs, such as those by Ivan Niven (1947), often involve integral calculus.

As mentioned in the classification section, $\pi$ and $e$ are also examples of transcendental numbers, a category of irrational numbers that are not roots of any non-zero polynomial equation with integer coefficients. Proving a number is transcendental is a stronger result than just proving it is irrational, and such proofs are typically very advanced.

Rationalisation of Denominators

Rationalizing the denominator is a process used to remove irrational numbers from the denominator of a fraction. The objective is to rewrite the fraction in an equivalent form where the denominator is a rational number. This practice simplifies expressions, makes calculations easier (especially before the widespread use of calculators), and is a standard way to present final answers in many mathematical contexts.

The process involves multiplying both the numerator and the denominator of the fraction by a carefully chosen term called a rationalizing factor. Since multiplying the numerator and denominator by the same non-zero value is equivalent to multiplying the entire fraction by 1, the value of the original fraction remains unchanged.

Methods Based on Denominator Form

The form of the irrational denominator dictates the choice of the rationalizing factor.

Case 1: The denominator is a single square root term, $\sqrt{a}$

If the denominator is of the form $b\sqrt{a}$ or simply $\sqrt{a}$, where $b$ is rational and $\sqrt{a}$ is irrational (meaning $a$ is a positive number that is not a perfect square of a rational number), the rationalizing factor is $\sqrt{a}$. Multiplying $\sqrt{a}$ by itself results in $a$, which is rational.

Multiply the numerator and the denominator by $\sqrt{a}$:

(If the denominator is just $\sqrt{a}$, then $b=1$). Since $b$ and $a$ are rational (assuming $a$ was initially rational), their product $ba$ is rational, and thus the denominator is rational.

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Case 2: The denominator is a binomial involving square roots (e.g., $a \pm \sqrt{b}$ or $\sqrt{a} \pm \sqrt{b}$)

If the denominator is a sum or difference of two terms, at least one of which is an irrational square root (and the overall expression is irrational), the rationalizing factor is the conjugate of the denominator. The conjugate of a binomial $(x+y)$ is $(x-y)$, and the conjugate of $(x-y)$ is $(x+y)$.

The product of a binomial and its conjugate is always a difference of squares: $(x+y)(x-y) = x^2 - y^2$. This is useful because squaring a square root term removes the radical (e.g., $(\sqrt{b})^2 = b$).

• If the denominator is $a + \sqrt{b}$, its conjugate is $a - \sqrt{b}$. Their product is $(a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b$. If $a$ and $b$ were rational, $a^2-b$ is rational.
• If the denominator is $\sqrt{a} + \sqrt{b}$, its conjugate is $\sqrt{a} - \sqrt{b}$. Their product is $(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = (\sqrt{a})^2 - (\sqrt{b})^2 = a - b$. If $a$ and $b$ were rational, $a-b$ is rational.

To rationalize, multiply the numerator and the denominator by the conjugate of the denominator.

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Rationalizing denominators of higher-order roots (like cube roots) or more complex irrational expressions follows the same principle: find a factor to multiply by that will make the denominator rational, often utilizing algebraic identities like those for sums/differences of cubes ($a^3 \pm b^3$) or higher powers.