Rational numbers — definition, decimal expansion (terminating - repeating)

Definition

Why include integers? Every integer $n$ can be written as $\frac{n}{1}$ , so integers are a subset of rationals: $\mathbb{Z} \subset \mathbb{Q}$ .

Why exclude $q = 0$ ? Division by zero is undefined in mathematics. If we allowed it, we'd break the field axioms (every non-zero element must have a multiplicative inverse).

Decimal Expansion: The Two Types

When we convert $\frac{p}{q}$ to decimal form using long division, exactly two things can happen:

Why this condition? Our decimal system is base 10, and $10 = 2 \times 5$ . When we perform long division, we repeatedly multiply the remainder by 10. If the denominator $q$ divides some power of 10exactly, the division terminates.

Derivation:

Start with $\frac{p}{q}$ in lowest terms
Assume the decimal terminates after $n$ digits
Then $\frac{p}{q} = \frac{k}{10^n}$ for some integer $k$
Cross-multiply: $p \cdot 10^n = k \cdot q$
Since gcd $(p, q) = 1$ , we need $q \mid 10^n$
Since $10^n = 2^n \cdot 5^n$ , the only prime factors of $10^n$ are 2 and 5
Therefore $q$ can only have prime factors 2 and 5

2. Non-Terminating Repeating Decimals

Why do they repeat? When performing long division, the possible remainders are limited: $0, 1, 2, .., q-1$ . Since we only have $q$ possible remainders, after at most $q$ steps, we must see a remainder we've seen before. Once a remainder repeats, the entire division pattern repeats.

Worked example Example 2: Repeating Decimal

Convert $\frac{5}{6}$ to decimal. Step 1: Check denominator $6 = 2 \cdot 3$ Contains 3(not 2 or 5) → Will repeat ✓

Step 2: Long division

5 ÷ 6:
 5.0000...
 remainder 5 → 50 ÷ 6 = 8 remainder 2
 remainder 2 → 20 ÷ 6 = 3 remainder 2  ← remainder 2 again!
 remainder 2 → 20 ÷ 6 = 3 remainder 2  ← pattern repeats

$\frac{5}{6} = 0.8333... = 0.8\overline{3}$ Why does 3 repeat? Once remainder 2 appears again (at step 2), we're back to dividing 20 by 6, which gives the same result forever.

Worked example Example 3: Longer Repetend

Convert $\frac{1}{7}$ to decimal. Check: $7 = 7^1$ (prime other than 2, 5) → Will repeat

Long division:

1 ÷ 7:
remainder 1 → 10 ÷ 7 = 1 remainder 3
remainder 3 → 30 ÷ 7 = 4 remainder 2
remainder 2 → 20 ÷ 7 = 2 remainder 6
remainder 6 → 60 ÷ 7 = 8 remainder 4
remainder 4 → 40 ÷ 7 = 5 remainder 5
remainder 5 → 50 ÷ 7 = 7 remainder 1  ← remainder 1 again!

$\frac{1}{7} = 0.\overline{142857}$

The repetend has length 6. Note: $\phi(7) = 6$ , confirming theory!

Converting Repeating Decimals Back to Fractions

Worked example Example 5: Mixed (non-pure) Repeating

Convert $0.58\overline{3} = 0.58333...$ to a fraction.

Step 1: Separate the non-repeating and repeating parts

Non-repeating: $0.58$ (2 digits)
Repeating: $\overline{3}$ (1 digit)

Step 2: Let $x = 0.58\overline{3}$ Multiply by $10^2 = 100$ to move past non-repeating: $100x = 58.\overline{3}$ Step 3: Multiply by $10^3 = 1000$ to align repetend: $1000x = 583.\overline{3}$ Step 4: Subtract: $1000x - 100x = 583.\overline{3} - 58.\overline{3}$ $900x = 525$ x = \frac{525}{900} = \frac{7}{12}$$

Why this method? We need two equations where the repeating part aligns perfectly for cancellation.

Why This Matters

Distinguishing Number Types: Decimal behavior immediately tells us if a number is rational or irrational
Computer Representation: Computers represent rationals exactly but approximate irrationals—understanding terminating vs. repeating helps predict precision issues
Foundation for Real Numbers: The rationals are dense in the reals (between any two real numbers, there's a rational), but they have "gaps" (the irrationals)
Practical Computation: Converting repeating decimals to fractions is essential in engineering calculations where exact values matter

Recall Explain to a 12-year-old

Imagine you have a pizza. A rational number is like saying "I ate3 out of 8 slices" — you can write it as a fraction: $\frac{3}{8}$ .

Now, when you divide 3 by 8 using a calculator or long division, you get $0.375$ . The decimal stops! That's called a terminating decimal. It happens when the bottom number (8) can be made using only 2's and 5's multiplied together. (8 = 2 × 2).

But what if you divide 1 by 3? You get $0.333333...$ — the 3's never stop! But they repeat forever. That's a repeating decimal. It happens when the bottom number has other numbers in it (like 3).

Here's the cool part: if you give me ANY decimal that either stops or repeats, I can convert it back into a fraction! But if the decimal goes on forever WITHOUT repeating (like $\pi$ ), then it's NOT a rational number — it's calledrrational, and you can't write it as a simple fraction.

So rational numbers are special: their decimals always play by the rules (stop or repeat). Irrational numbers are rebels!

Connections

Prime factorization — essential for checking terminating condition
Euler's totient function — determines maximum repetend length
Real numbers — rationals are a proper subset of reals
Irrational numbers — numbers whose decimals don't terminate or repeat
Dense sets — rationals are dense in the reals
Long division algorithm — the computational process revealing decimal patterns
Modular arithmetic — explains why remainders cycle in repeating decimals

Flashcards

What is a rational number? :: A number that can be expressed as $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$ .

When does a rational number have a terminating decimal?

When the denominator (in lowest terms) has the form

q = 2^a \cdot 5^b

— only prime factors 2 and 5.

When does a rational number have a repeating decimal?

When the denominator (in lowest terms) contains at least one prime factor other than 2 or 5.

Why do repeating decimals repeat?

In long division by

q

, there are only

q

possible remainders. Eventually a remainder must repeat, causing the division pattern to repeat.

How to convert

0.\overline{abc}

to a fraction?

Let

x = 0.\overline{abc}

, multiply by

10^n

(where

n

is repetend length), subtract original, solve for

x

. Result:

x = \frac{abc}{10^n - 1}

What is

\frac{1}{8}

in decimal?

0.125

(terminates because

8 = 2^3

)

What is

\frac{1}{6}

in decimal?

0.1\overline{6}

(repeats because

6 = 2 \cdot 3

contains prime3)

True or False: All decimals are rational numbers.

False. Only terminating or repeating decimals are rational. Non-repeating infinite decimals are irrational.

What is

0.\overline{9}

Exactly1. Proof: Let

x = 0.\overline{9}

, then

10x = 9.\overline{9}

, so

10x - x = 9

, giving

x = 1

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Rational numbers ka matlab hai "ratio" wale numbers – yani jo numbers hum fraction ke form mein likh sakte hain, jaise $\frac{3}{4}$ ya $\frac{-7}{2}$ . Yeh definition simple lagti hai, lekin iska sabse interesting part hai inke decimal expansion mein – jab hum fraction ko decimal mein convert karte hain, tab hamesha do hi tarah ki patterns dekhne ko milti hain.

Pehli type: Terminating decimals – jaise $\frac{7}{8} = 0.875$ . Yeh decimals khatam ho jate hain, infinite nahi jate. Yeh tab hota hai jab denominator (neeche wala number) sirf 2 aur 5 ke powers se bana ho, kyunki hamara decimal system base-10 hai ( $10 = 2 \times 5$ ). Jab denominator 10 kisi power se perfectly divide ho jata hai, tab division process stop ho jaata hai.

Dosri type: Repeating decimals – jaise $\frac{5}{6} = 0.8333...$ jisme 3 infinite times repeat hota hai. Yeh tab hota hai jab denominator mein 2 aur 5 ke alawa koi aur prime factor ho (jaise 3, 7, 11). Long division karte waqt remainders limited hote hain (sirf 0 se q-1 tak), toh eventually koi remainder repeat hona hi hai – aur jab remainder repeat ho, toh pura pattern repeat hone lagta hai.

Yeh kyun important hai? Isse hum turant pata laga sakte hain ki koi number rational hai ya irrational.Agar decimal terminate ya repeat karta hai, toh rational hai. Agar decimal infinite jaye BINA repeat kiye (jaise $\pi$ ya $\sqrt{2}$ ), toh irrational hai. Engineering aur science mein jab exact calculations chahiye, tab repeating decimals ko wapas fractions mein convert karna zaroori hota hai – aur iska ek fixed mathematical method hai (multiply by $10^n$ , subtract, solve). Yeh samajhna number theory ki foundation hai!

Rational numbers — definition, decimal expansion (terminating - repeating)

Definition

Decimal Expansion: The Two Types

1. Terminating Decimals

2. Non-Terminating Repeating Decimals

Converting Repeating Decimals Back to Fractions

Why This Matters

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Number Theory (Intermediate)

Connections