Real Numbers & Decimal Expansions

Imagine the number line as a packed cricket stadium where every single seat is filled — no empty gaps anywhere. Those seats are the real numbers. This lesson explains what real numbers are, how the rationals and irrationals combine to fill the entire line, and why the perfect match between points and numbers is such a powerful idea.

Definition: A real number is any number that can be placed on the number line. The set of all real numbers is written R.

Definition: The real number line is a straight line on which every point corresponds to exactly one real number, and every real number corresponds to exactly one point.

Rationals + irrationals = the reals

You have met two families of numbers. The rationals can be written as p/q (fractions, integers, terminating and recurring decimals). The irrationals cannot — their decimals run forever without repeating (√2, π, and so on). Put these two families together and you get the complete real number system:

R = rationals ∪ irrationals

Crucially, these two families do not overlap (no number is both rational and irrational) and together they leave nothing out. Every number you will normally meet — whole numbers, fractions, negative numbers, square roots, π — is real.

Why it matters: Before irrationals were included, the number line looked full but secretly had holes — a point like √2 had no number to name it. Adding irrationals plugs every hole. This property, called completeness, is what makes calculus, measurement and physics possible: any length, area, time or temperature you can imagine has an exact real-number value.

The one-to-one correspondence

The deepest idea here is a perfect pairing: every point on the line is some real number, and every real number sits at some point on the line. There are no leftover points and no leftover numbers. Mathematicians say the real numbers and the points of the line are in one-to-one correspondence.

This is why the number line is such a faithful picture of arithmetic. Moving right means getting larger, moving left means getting smaller, distance between two points equals the difference of their values, and "between" on the line means "between" in size. Nothing is lost in translation.

How the number families nest

The real numbers sit at the top of a nested family of number systems you have built up over the years. Each set is contained in the next:

Natural numbers (1, 2, 3, …) ⊂ Whole numbers (0, 1, 2, …) ⊂ Integers (…, −2, −1, 0, 1, 2, …) ⊂ Rationals (p/q) ⊂ Real numbers (R).

The irrationals branch off from the rationals — they are real, but not rational. Together rationals and irrationals reunite to form R.

Question: Classify each number as rational or irrational, and confirm all are real: 7, −3/4, 0.272727…, √16, √7, π.
Solution:
Step 1: 7 is an integer, hence rational.
Step 2: −3/4 is already p/q, rational.
Step 3: 0.272727… is a recurring decimal, so rational (it equals 27/99 = 3/11).
Step 4: √16 = 4 (perfect square), rational.
Step 5: √7 is the root of a non-perfect-square, so irrational.
Step 6: π has a non-terminating, non-repeating decimal, so irrational.
Conclusion: Rational: 7, −3/4, 0.2727…, √16. Irrational: √7, π. All six are real numbers, since every one of them sits on the number line.

Real-world example: Any physical measurement you take — the temperature outside (28.6 °C), your height (1.62 m), the distance to school (3.4 km), or the diagonal of a square tile (√2 m) — is a real number. The real number system is precisely the toolkit science needs to express any quantity on a continuous scale.

Common misconception: "Real numbers are a brand-new kind of number, separate from fractions and roots." No — real numbers are the umbrella that already contains every rational and every irrational you know. There is no separate "real-only" number.

Common misconception: "There are gaps between numbers on the line." There are no gaps. Between any two reals there are infinitely many more reals, both rational and irrational. The line is continuous and complete.

Common misconception: "Imaginary numbers like √−1 are real numbers." They are not. √−1 has no point on the ordinary number line, so it is not a real number — it belongs to a larger system (complex numbers) studied later.

Type 1 ÷ 8 into a calculator and the answer stops cleanly at 0.125. Type 1 ÷ 3 and the 3s loop forever. Why do some fractions end and others run on? This lesson reveals the simple rule hiding behind that behaviour, shows you how to predict it without dividing, and clears up the most common confusion: that a repeating decimal is somehow "not a real fraction."

Definition: A terminating decimal ends after a finite number of digits (e.g. 0.75, 0.125).

Definition: A non-terminating recurring decimal never ends, but a block of digits repeats forever (e.g. 0.333…, 0.142857142857…). The repeating block is shown with a bar, e.g. 0.3̄.

Every rational has one of exactly two decimal forms

This is the key fact: when you express any rational number p/q as a decimal, the result is always either terminating or non-terminating recurring — never anything else. A rational number can never produce a non-repeating endless decimal; that behaviour belongs only to irrationals.

Why only two outcomes? When you do long division by q, the remainder at each step must be one of the numbers 0, 1, 2, …, q − 1 — only q possible values. Either a remainder of 0 appears (the division stops — terminating) or, within at most q steps, a remainder must repeat. Once a remainder repeats, the whole digit pattern from that point repeats too, forever — giving a recurring decimal. There is simply no third possibility.

Why it matters: This tells you that "terminating or recurring" is the signature of a rational number. If you ever see a decimal that goes on forever without any repeating block, you are looking at an irrational number, not a fraction.

The deciding rule: look at the denominator's prime factors

You can predict which type you will get without dividing at all, just by factorising the denominator (in lowest terms):

A fraction in lowest terms gives a terminating decimal if and only if its denominator's only prime factors are 2 and/or 5. If the denominator has any other prime factor (3, 7, 11, …), the decimal recurs.

The reason is that our decimal system is base 10, and 10 = 2 × 5. A denominator built only from 2s and 5s can be scaled up to a power of 10 (like 10, 100, 1000), which is exactly what a terminating decimal is. Any other prime factor cannot be turned into a power of 10, so the division never reaches a remainder of 0.

Question: Without dividing, decide whether 7/40 and 5/12 terminate or recur.
Solution:
Step 1: Factorise the denominator of 7/40. 40 = 2 × 2 × 2 × 5 = 2³ × 5 — only 2s and 5s.
Step 2: Since 40 has no other prime factor, 7/40 terminates. (Indeed 7/40 = 0.175.)
Step 3: Factorise the denominator of 5/12. 12 = 2 × 2 × 3 = 2² × 3 — it contains a 3.
Step 4: Because of the prime factor 3, 5/12 recurs. (Indeed 5/12 = 0.41666… = 0.41̄6̄.)
Conclusion: 7/40 terminates; 5/12 is non-terminating recurring.

Always reduce to lowest terms first. For example 15/24 looks like it has a 3, but 15/24 = 5/8, whose denominator is 2³ — so it actually terminates (0.625).

Recurring decimals are still rational

A recurring decimal is fully a rational number — it can always be converted back to p/q. For example 0.333… = 1/3 and 0.1818… = 2/11. So do not be fooled by the endless tail: an endless decimal is irrational only if it also never repeats.

Real-world example: Money usually terminates — a price of ₹0.50 is exactly 50 paise. But split a ₹100 restaurant bill three ways and each share is ₹33.333…, a recurring decimal. In practice you round to ₹33.33 and someone covers the extra paisa — the recurring tail is why three-way splits never come out perfectly even.

Common misconception: "A recurring decimal is irrational because it never ends." Wrong — recurring decimals are rational. The test for irrationality is non-terminating and non-repeating together.

Common misconception: "You have to divide to know whether a fraction terminates." You don't — just factorise the denominator (in lowest terms) and check for prime factors other than 2 and 5.

Common misconception: "1/6 terminates because 6 has a 2 in it." It must be only 2s and 5s. 6 = 2 × 3, and that stray 3 forces 1/6 = 0.1666… to recur.

Recurring decimals look intimidating, but they are secretly fractions wearing a disguise. With one clever algebra trick you can unmask any of them and recover the exact p/q form. This lesson teaches that method step by step, extends it to multi-digit and mixed repeats, and uses it to settle the famous "0.999… = 1" debate.

Definition: A recurring (repeating) decimal is a non-terminating decimal in which a block of digits repeats forever, written with a bar over the block — e.g. 0.6̄ = 0.6666…, 0.2̄7̄ = 0.272727…

Definition: Converting to p/q form means writing the decimal as a fraction of two integers, which proves it is rational.

Why any recurring decimal is rational

A repeating decimal never ends, yet it equals an exact fraction. The reason is that the endless repeating tail can be cancelled against itself. If you multiply the number by the right power of 10, you shift the decimal point so the repeating tails of the two versions line up perfectly. Subtracting then wipes out the infinite tail, leaving an ordinary equation you can solve. The number of repeating digits tells you which power of 10 to use: one repeating digit → multiply by 10, two → by 100, three → by 1000, and so on.

Why it matters: This proves that "non-terminating recurring" decimals are firmly rational, and gives you a guaranteed procedure to find the fraction — a frequent exam task. It also resolves paradoxes like 0.999… = 1 with a clean, convincing argument.

The method on a single-digit repeat

Question: Convert 0.6̄ (= 0.6666…) to a fraction.
Solution:
Step 1: Let x = 0.6666…
Step 2: One digit repeats, so multiply both sides by 10: 10x = 6.6666…
Step 3: Subtract the first equation from the second: 10x − x = 6.6666… − 0.6666…, which gives 9x = 6 (the endless tails cancel exactly).
Step 4: Solve: x = 6/9 = 2/3.
Conclusion: 0.6̄ = 2/3.

When more than one digit repeats

Match the power of 10 to the length of the repeating block.

Question: Convert 0.2̄7̄ (= 0.272727…) to a fraction.
Solution:
Step 1: Let x = 0.272727…
Step 2: Two digits repeat, so multiply by 100: 100x = 27.272727…
Step 3: Subtract: 100x − x = 27.272727… − 0.272727…, giving 99x = 27.
Step 4: Solve: x = 27/99 = 3/11.
Conclusion: 0.2̄7̄ = 3/11.

Notice the pattern: one repeating digit gives a denominator of 9, two give 99, three give 999, and so on — a useful shortcut for purely repeating decimals.

Mixed decimals (some non-repeating digits first)

When digits sit before the repeating block, multiply by two powers of 10 — one to clear the non-repeating part and one to shift a full repeat — then subtract.

Question: Convert 0.16̄ (= 0.16666…, where only the 6 repeats) to a fraction.
Solution:
Step 1: Let x = 0.16666…
Step 2: There is 1 non-repeating digit, so first multiply by 10: 10x = 1.6666…
Step 3: One digit repeats, so also multiply by 100: 100x = 16.6666…
Step 4: Subtract the two new equations: 100x − 10x = 16.6666… − 1.6666…, giving 90x = 15.
Step 5: Solve: x = 15/90 = 1/6.
Conclusion: 0.16̄ = 1/6.

Real-world example: This method settles the classic mind-bender that 0.999… = 1. Let x = 0.999…; then 10x = 9.999…; subtracting gives 9x = 9, so x = 1 exactly. So 0.999… is not "just below" 1 — it is another name for 1, just as 2/2 is another name for 1.

Common misconception: "0.999… is a tiny bit less than 1." There is no gap — the algebra forces x = 1 exactly. 0.999… and 1 are two names for the same number.

Common misconception: "Recurring decimals are irrational." The whole point of this method is that every recurring decimal converts to p/q, so it is rational. Only non-repeating endless decimals are irrational.

Common misconception: "Always multiply by 10." The power of 10 must match the length of the repeating block: 10 for one repeating digit, 100 for two, 1000 for three. Use 10 blindly on a two-digit repeat and the tails won't cancel.

Every number you will ever meet in school maths — whole, fraction, root, or π — is a real number, and each one hides its true identity in its decimal expansion. This lesson teaches you to read that decimal like a detective: terminating, recurring, or neither, and to convert any repeating decimal back into a fraction.

Definition: The real numbers are all the rationals together with all the irrationals. Each real number corresponds to exactly one point on the number line, and each point corresponds to exactly one real number — a perfect, gap-free one-to-one match.

The number line is complete

The rationals alone leave "gaps" (like the spot for √2). Filling in the irrationals plugs every gap. The result: the number line and the set of real numbers are in perfect one-to-one correspondence. Pick any point — there is exactly one real number there; pick any real number — there is exactly one point for it. This completeness is what makes the real line a continuous, unbroken thread.

Why it matters: Because of this match, we can see numbers as positions and measure positions as numbers. Geometry and arithmetic become two views of the same thing.

Reading a number from its decimal — three cases

The decimal expansion classifies any real number instantly.

Case 1 — Terminating decimal. The division ends. Example: 7/8 = 0.875. After three places it stops. Terminating decimals are always rational.

Case 2 — Non-terminating but recurring. The decimal never ends but a block of digits repeats forever. Examples: 1/3 = 0.333… (written 0.3̅, a bar over the 3) and 1/7 = 0.142857142857… (the block 142857 repeats). These are also rational — repetition is the signature of a fraction.

Case 3 — Non-terminating and non-recurring. The decimal never ends and never settles into a repeating block. Example: 0.1010010001… (the number of zeros keeps growing, so no block repeats). These are irrational.

Why it matters: A surprising truth is that every fraction p/q gives either a terminating or a recurring decimal — never anything else. So if you ever see a decimal that genuinely never repeats, you are looking at an irrational number.

Converting a recurring decimal to p/q

Recurring decimals are rational, so they must equal some fraction. Here is the standard method.

Method: Let x equal the decimal. Multiply x by a power of 10 large enough to shift the decimal point past one full repeating period. Subtract the original from this, so the endless repeating tails cancel exactly, leaving a clean equation you can solve.

Question: Convert 0.3̅ (i.e. 0.333…) to a fraction.
Solution:
Step 1: Let x = 0.333…
Step 2: The repeating block is one digit, so multiply by 10: 10x = 3.333…
Step 3: Subtract the first from the second; the 0.333… tails cancel:
10x − x = 3.333… − 0.333… → 9x = 3.
Step 4: Solve: x = 3/9 = 1/3.
Conclusion: 0.3̅ = 1/3. ✓

Why subtraction works: both 10x and x have the same infinite repeating tail, so subtracting wipes the tail out completely and leaves a finite number — that is the whole reason we multiply by exactly one period's worth of 10s.

Locating a decimal by successive magnification

To pin down, say, 3.76 on the number line, we "zoom in" in stages. First find it between 3 and 4; then magnify the interval 3.7 to 3.8 and locate 3.76 inside it. Each zoom narrows the search by a factor of ten. This visual process shows that every decimal, no matter how many places, has a definite home on the line.

Real-world example: A digital weighing scale reading 3.76 kg has "found" the exact point between 3.7 and 3.8 — the same zoom-in logic the magnification picture shows.

Common misconception: "1/7 has a short repeat like 0.14." Wrong — its period is six digits long: 0.142857142857…. Always carry the long division until a remainder repeats; only then have you found the full repeating block.

Common misconception: "Recurring decimals aren't 'real' fractions." They are exactly fractions — 0.3̅ is 1/3, not merely close to it.

A decimal like 0.4777… looks like it could ramble on forever with no pattern — but the moment you spot that only the 7 repeats, you hold the key to turning it back into an exact fraction. The algebra trick that does this is one of the most satisfying in the whole Number Systems chapter.

Definition: A recurring (or repeating) decimal is a decimal in which, after some point, a block of digits repeats forever. We mark the repeating block with a bar: 0.4̄7 means 0.4777… where only the 7 repeats. Every recurring decimal is a rational number — it can always be written as p/q.

Definition: A mixed recurring decimal has some non-repeating digit(s) right after the decimal point, followed by a repeating block. Here the 4 is non-repeating and the 7 is the repeating block.

The core idea: line up the tails so they cancel

The whole technique rests on one observation: if two numbers have the exact same infinite repeating tail, subtracting one from the other makes that endless tail vanish, leaving a clean finite number. Our job is to create two such numbers by multiplying x by suitable powers of 10, then subtract.

There are two "shifts" to handle in a mixed recurring decimal:

1. Shift past the non-repeating part (the 4) — multiply so the decimal point sits just before the first repeating digit.
2. Shift one full repeating period — multiply again by 10 raised to (number of repeating digits) so a second copy lines up.

Worked example

Question: Express 0.4777… (only the 7 repeats, written 0.4̄7) in the form p/q.

Solution:

Step 1: Let x = 0.4777…

Step 2: One digit (the 4) does not repeat, so first shift past it by multiplying by 10:
10x = 4.777… …(i)
Now the decimal point sits immediately before the repeating 7s, with the pure recurring tail .777… exposed.

Step 3: The repeating block has one digit, so multiply (i) by 10 once more to slide along exactly one full period:
100x = 47.777… …(ii)
Notice (i) and (ii) now have the identical tail .777….

Step 4: Subtract (i) from (ii) so the matching repeating tails cancel:
100x − 10x = 47.777… − 4.777…
90x = 43

Step 5: Solve for x:
x = 43/90

Check: Dividing 43 by 90 gives 0.4777…, which matches the original. So 0.4̄7 = 43/90.

Note on lowest terms: 90 = 2 × 3 × 3 × 5, while 43 is prime, so they share no common factor. Hence 43/90 is already in lowest terms.

Reading the answer as a shortcut

There is a pattern worth memorising for mixed recurring decimals:

x = (whole number formed by ALL digits up to the end of the first repeat) − (number formed by the NON-repeating digits), divided by (as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits).

Applying it to 0.4̄7: the digits up to the end of the first repeat are "47", the non-repeating digits are "4", there is 1 repeating digit (→ one 9) and 1 non-repeating digit (→ one 0). So
x = (47 − 4) / 90 = 43/90. ✓
This matches the algebra exactly — the algebra is the reason the shortcut works.

Why it matters: Converting decimals to fractions is essential for exact computation. In exams, marks are awarded for the method (let x …, the two multiplications, the subtraction), not just the final fraction. The skill also proves a deeper claim: that every terminating or recurring decimal is rational, which is one half of the fundamental link between decimals and rational numbers.

Real-world example: Calculators and computers store many fractions as recurring decimals (1/3 = 0.333…, 1/7 = 0.142857…). Reversing the process — recovering the exact fraction — is exactly how you avoid rounding errors when, say, totalling repeated measurements or splitting a bill into thirds.

Common misconceptions corrected

Common misconception: "Just multiply by 100 straight away." If you skip Step 2 and only do 100x and x, the tails will not align (because of the non-repeating 4) and the subtraction leaves a fractional remainder. You must first shift past the non-repeating digits.

Common misconception: "0.4777… equals 0.48 or roughly 47/100." No — the 7s continue forever; it is exactly 43/90 ≈ 0.4777…, not a terminating decimal.

Common misconception: "Multiply by 10 as many times as total digits shown." The multipliers are governed by the structure (non-repeating count and repeating-block length), not by how many digits you happened to write out.

When a decimal repeats a two-digit block like 1.272727…, the same cancel-the-tail trick still works — you just have to jump two digits at a time instead of one. Master this and you can convert any pure recurring decimal into an exact fraction in seconds.

Definition: A pure recurring decimal is a decimal whose repetition begins immediately after the decimal point, with no non-repeating digits in between. Here, 1.2̄7̄ means 1.272727…, where the whole block 27 repeats forever. (The integer part 1 simply rides along and is added back at the end — the recurring action is in the decimal part.)

Why the multiplier is 100, not 10

The governing rule for the "period shift" is: multiply by 10 raised to the number of repeating digits. The repeating block here is 27, which has two digits, so we multiply by 10² = 100. This shifts the decimal point exactly one full period to the right, producing a second number with the identical infinite tail — the precondition for the tails to cancel on subtraction.

If you mistakenly used 10, you would shift only half a period (127.2727… vs 1.2727…), the tails would not match, and the subtraction would leave behind an unwanted recurring remainder. The block length dictates the power of 10 — always.

Worked example

Question: Express 1.272727… (the block 27 repeats, written 1.2̄7̄) in the form p/q.

Solution:

Step 1: Let x = 1.272727…

Step 2: The repeating block has TWO digits (27), so multiply by 100 to shift exactly one full period:
100x = 127.272727… …(i)

Step 3: Keep the original equation:
x = 1.272727… …(ii)

Step 4: Subtract (ii) from (i); the identical recurring tails .272727… cancel perfectly:
100x − x = 127.272727… − 1.272727…
99x = 126

Step 5: Solve:
x = 126/99

Step 6: Simplify to lowest terms. The HCF of 126 and 99 is 9 (126 = 2 × 3² × 7; 99 = 3² × 11; common part 3² = 9):
126 ÷ 9 = 14, and 99 ÷ 9 = 11, so x = 14/11.

Check: 14 ÷ 11 = 1.2727…, confirming the answer. Thus 1.2̄7̄ = 14/11.

The pattern behind it (pure recurring shortcut)

For a pure recurring decimal, the fraction is simply:
(the repeating block) divided by (as many 9s as there are digits in the block).

For the decimal part 0.2̄7̄ this gives 27/99 = 3/11 (dividing top and bottom by 9). Adding the integer part: 1 + 3/11 = 14/11. ✓ This is exactly why "99" appeared in the denominator: two repeating digits → two 9s. One repeating digit would give a single 9 (e.g. 0.3̄ = 3/9 = 1/3); three would give 999, and so on.

Why it matters: This confirms that every recurring decimal is rational — no exceptions. It also gives you a fast mental check: 0.9̄9̄… type denominators (9, 99, 999) instantly signal pure recurring decimals. In exams, showing the "let x …, ×100, subtract" method earns full method marks, and simplifying to 14/11 earns the final mark.

Real-world example: Many everyday fractions produce two-digit recurring blocks: 1/11 = 0.090909… = 0.0̄9̄, 4/33 = 0.121212…, and indeed 14/11 = 1.2727…. Recognising the block length lets you instantly reverse such a calculator readout back into the clean fraction.

Common misconceptions corrected

Common misconception: "Multiply by 10 because there's a decimal point." The multiplier depends on the length of the repeating block (here 2 digits → ×100), not on the mere presence of a decimal point.

Common misconception: "126/99 is the final answer." It is correct but not in lowest terms. Always divide by the HCF (9 here) to reach 14/11. Exam answers are expected in simplest form.

Common misconception: "The leading 1 changes the method." It does not — the integer part is carried through unchanged; only the recurring fractional part drives the cancellation, and you re-attach the integer in the final value (it is already included when you set x = 1.2727…).

Every real number wears a "decimal fingerprint" — and just by glancing at how that decimal behaves, you can instantly tell whether the number is rational or irrational. This single classification ties together everything in the Number Systems chapter.

Definition: A rational number can be written as p/q with integers p, q and q ≠ 0. Its decimal expansion is always either terminating or non-terminating recurring.

Definition: An irrational number cannot be written as p/q. Its decimal expansion is non-terminating and non-recurring (endless, with no repeating block).

The three kinds of decimal expansion

Sort any real number's decimal into exactly one of three boxes:

• Terminating (the decimal stops) ⟹ rational. Example: 0.875 = 7/8. The division simply ends.
• Non-terminating recurring (goes forever but a block repeats) ⟹ rational. Example: 0.3̄ = 0.333… = 1/3. It never stops, yet it is still a ratio of integers.
• Non-terminating non-recurring (goes forever with no repeating block) ⟹ irrational. Example: 0.1010010001000010… (the gaps of 0s keep growing) — and famously √2 = 1.41421356… and π = 3.14159….

Why it matters: This gives you a foolproof test. You never have to guess: examine the decimal, and the box it lands in tells you the number's family. It also reveals a deep truth — terminating and recurring decimals together account for all and only the rational numbers; anything that escapes both is irrational.

Common misconception: "Non-terminating always means irrational." False — 0.333… never terminates yet is the rational 1/3. The deciding factor for non-terminating decimals is whether they recur.

Shortcut 1 — purely recurring decimal to fraction

If the repetition begins right after the decimal point, use:
a block of n repeating digits equals (the block) ÷ (n nines).

Example: 0.2̄7̄ = 27/99 = 3/11. The block "27" has 2 digits, so divide by 99; then simplify (÷9). Likewise 0.3̄ = 3/9 = 1/3, and 0.1̄2̄3̄ = 123/999.

Shortcut 2 — mixed recurring decimal to fraction

If some non-repeating digit(s) come before the repeating block, use:
x = [ (whole number from ALL digits up to the end of the first repeat) − (number from the NON-repeating digits) ] ÷ [ (as many 9s as repeating digits) followed by (as many 0s as non-repeating digits) ].

Example: 0.4̄7 (the 4 is non-repeating, the 7 repeats).

• Digits up to end of first repeat: 47. Non-repeating digits: 4. Repeating digits: 1 (→ one 9). Non-repeating digits: 1 (→ one 0).
• x = (47 − 4) / 90 = 43/90.

Worked example.
Question: Express 0.5̄ as a fraction and state its type.
Solution:
Step 1: Pure recurring, block "5", 1 digit ⟹ denominator 9.
Step 2: 0.5̄ = 5/9.
Conclusion: 0.5̄ = 5/9, a rational number (non-terminating recurring).

Worked example.
Question: Express 0.2̄3̄ as a fraction in lowest terms.
Solution:
Step 1: Pure recurring, block "23", 2 digits ⟹ denominator 99.
Step 2: 0.2̄3̄ = 23/99.
Step 3: 23 is prime and does not divide 99, so it is already in lowest terms.
Conclusion: 0.2̄3̄ = 23/99.

The number line correspondence

Definition (completeness): Every real number corresponds to a unique point on the number line, and every point on the number line corresponds to a unique real number. This one-to-one matching is why the number line has no gaps: rationals alone would leave holes at √2, π, etc.; the irrationals plug exactly those holes. Together, rationals and irrationals (the real numbers) fill the line completely.

Real-world example: A ruler shows this completeness physically — between any two marks (say 1 cm and 2 cm) lie infinitely many rationals (1.5, 1.25, …) and infinitely many irrationals (1 + 1/√2, …). Any precise length you measure sits at exactly one real point, rational or irrational.

Common misconception: "There are more rationals than irrationals because we see fractions everywhere." In fact the line is densely packed with both, and the irrationals are if anything the more numerous; both are infinite and fill the line jointly.

Every number you have ever used — whether counting marbles, measuring length, or finding the diagonal of a square — lives somewhere on a single, unbroken line. This lesson pulls together the big ideas about real numbers and how their decimal expansions reveal whether a number is rational or irrational, and shows you the exact techniques to convert recurring decimals back into fractions.

Definition: A real number is any number that corresponds to a point on the number line — this includes both rational numbers (which can be written as p/q with integers p, q and q ≠ 0) and irrational numbers (which cannot).

Real numbers fill the number line completely

The deepest fact about real numbers is that there is a perfect one-to-one correspondence between points on the number line and real numbers. Pick any point — there is exactly one real number for it; pick any real number — there is exactly one point for it. There are no "gaps." If you only had the rational numbers, the line would be full of holes (for instance, at the point whose distance from 0 equals √2). The irrationals plug every one of those holes.

Why it matters: This is what makes the number line a faithful picture of arithmetic. When you draw a length, find an area, or read a thermometer, you can always name the result by a real number — never anything "off the line."

Rational numbers: decimals that terminate or repeat

When you carry out the long division p ÷ q, only finitely many remainders are possible (the remainders must be less than q). So eventually either the remainder becomes 0 (the decimal terminates, like 1/4 = 0.25) or a remainder repeats, forcing a block of digits to repeat forever (the decimal is non-terminating recurring, like 1/3 = 0.333… or 1/7 = 0.142857142857…).

Definition: A recurring (or repeating) decimal is one in which a fixed block of digits repeats endlessly; we mark the repeating block with a bar, e.g. 0.3̄ or 0.1̄42857̄.

Real-world example: A baker splitting a 1 kg cake equally among 3 people gives each 0.333… kg — a perfectly real, exact amount, even though its decimal never ends.

Irrational numbers: decimals that never end and never repeat

Definition: An irrational number has a decimal expansion that is non-terminating and non-recurring — it goes on forever with no repeating block.

Examples include √2 = 1.41421356…, √3 = 1.73205…, and π = 3.14159265…. Because there is no repeating pattern, no matter how far you compute, you can never write them as a clean fraction p/q.

Common misconception: "π = 22/7 exactly." Wrong — 22/7 is only a convenient approximation. 22/7 = 3.142857142857… (recurring, hence rational), but π is irrational; they differ from the third decimal place onward.

Converting a recurring decimal to p/q

The general engine: set x equal to the decimal, multiply by suitable powers of 10 so the repeating tails line up, then subtract to cancel the infinite tail.

Question: Express 0.3̄ as a fraction.
Solution:
Step 1: Let x = 0.333…
Step 2: Multiply by 10: 10x = 3.333…
Step 3: Subtract: 10x − x = 3.333… − 0.333… → 9x = 3.
Conclusion: x = 3/9 = 1/3.

Question: Express 1.2̄7̄ (i.e. 1.272727…) as a fraction.
Solution:
Step 1: Let x = 1.272727…
Step 2: Two digits repeat, so multiply by 100: 100x = 127.2727…
Step 3: Subtract: 100x − x = 127.2727… − 1.2727… → 99x = 126.
Conclusion: x = 126/99 = 14/11.

Handy shortcuts for purely and mixed recurring decimals

• Purely recurring: 0.a̅b̅ = ab/99, 0.a̅b̅c̅ = abc/999 (one 9 per repeating digit). Example: 0.4̄5̄ = 45/99 = 5/11.
• Mixed recurring (some non-repeating digits before the bar): 0.a b̄ = (ab − a)/90. Example: 0.16̄ = (16 − 1)/90 = 15/90 = 1/6.

Why it matters: These let you instantly check answers and convert decimals without re-deriving the subtraction each time — useful under exam time pressure.

Successive magnification: locating a decimal on the line

To place a number like 2.665 precisely, "zoom in." First locate it between 2 and 3; divide that gap into ten parts to find it between 2.6 and 2.7; divide again to find it between 2.66 and 2.67; once more between 2.665 and 2.666. This process of successive magnification can pin down a real number to as many decimal places as you like.

Common misconception: "1/7 just repeats the digit 1, giving 0.111…." Wrong — that's 1/9. For 1/7 you must keep dividing until a remainder repeats; the actual period has length 6: 0.142857142857…. Always finish the long division.

Cover the answer, recall, then check. 7 cards on real numbers and decimals.

Q1. What are real numbers?
A1. All rational and irrational numbers together form the real numbers.

Q2. What kind of decimal expansion does a rational number have?
A2. Either terminating or non-terminating but recurring (repeating).

Q3. What kind of decimal expansion does an irrational number have?
A3. Non-terminating and non-recurring.

Q4. Express 0.3̄ (0.333...) as a fraction.
A4. Let x = 0.333...; 10x = 3.333...; subtracting, 9x = 3, so x = 1/3.

Q5. Every point on the number line represents what kind of number?
A5. A unique real number.

Q6. Is every real number either rational or irrational?
A6. Yes — the real numbers are made up entirely of rationals and irrationals.

Q7. When does a rational number p/q have a terminating decimal expansion?
A7. When the denominator q (in lowest terms) has only 2 and/or 5 as prime factors.