Number and Algebraic Foundations
The Real Number System and Its Subsets
How the counting numbers, integers, rationals, reals and complex numbers nest inside one another, why every recurring decimal is a fraction, and how to convert one.
What this page covers
- State the five standard number sets and the containment chain between them
- Decide which set a given number belongs to, and justify the answer
- Convert any recurring decimal to an exact fraction using the shift-and-subtract method
- Recognise why the reals must be extended to the complex numbers
Why the number sets are worth naming
Almost every rule in algebra carries an unstated condition about which numbers are allowed. 'Every number has a reciprocal' is false over the integers and true over the non-zero rationals. 'Every equation x2 = a has a solution' is false over the reals and true over the complex numbers. Naming the sets makes those conditions visible instead of leaving them to be discovered by accident.
The five sets below nest inside one another. Each is built from the one before it by adding whatever was missing, and each extension is forced by an equation the earlier set could not solve.
| Set | Symbol | Members | Added to solve |
|---|---|---|---|
| Counting (natural) numbers | ℕ | 1, 2, 3, … | — the starting point |
| Integers | ℤ | …, -3, -2, -1, 0, 1, 2, 3, … | x + 3 = 1 |
| Rational numbers | ℚ | all p/q with p, q integers and q ≠ 0 | 3x = 1 |
| Real numbers | ℝ | every point on the number line, including π, √2, 3√7 | x2 = 2 |
| Complex numbers | ℂ | all a + bi with a, b real and i2 = -1 | x2 = -9 |
Read left to right: every counting number is an integer, every integer is a rational, every rational is a real, and every real is a complex number with zero imaginary part. None of the reverse statements holds.
The containments are strict, and it is worth being able to name a witness for each one. -3 is an integer but not a counting number. 1/2 is rational but not an integer. √2 is real but not rational. 3i is complex but not real. A student who can produce those four witnesses on demand understands the chain; one who can only recite it does not.
Integers: closing addition under subtraction
Within the counting numbers, x + 3 = 1 has no solution. Subtraction leads out of the set. The integers repair this by adjoining zero and the negatives, and the repair is exactly what is needed: subtraction of any integer from any integer lands back inside ℤ.
The integers are evenly spaced and isolated. Between any two neighbours there is nothing else in ℤ — which is precisely what the rationals will change.
A set is called closed under an operation when the operation never produces a result outside the set. ℕ is closed under addition and multiplication but not subtraction. ℤ is closed under addition, subtraction and multiplication but not division. Each extension in the chain closes one more operation.
Rational numbers and the recurring decimal
A rational number is any number expressible as pq with p and q integers and q ≠ 0. The condition q ≠ 0 is not decoration: division by zero is undefined, and a large share of algebraic mistakes come from losing track of it.
Every rational number has a decimal expansion that either terminates or eventually repeats a fixed block forever. The converse also holds, and it is the more useful direction: any decimal with a repeating block is a fraction, and there is a mechanical way to find it.
Let a be the recurring decimal and let the repeating block have length n. Multiply a by 10n so that the two numbers have identical tails, then subtract. The tails cancel exactly and a finite equation remains.
Worked example — converting 0.454545…
The repeating block is 45, of length 2, so multiply by 102 = 100.
The subtraction works because both numbers have the same infinite tail from the decimal point onward. Nothing is approximated; the equality is exact.
5 ÷ 11 = 0.454545… Divide 5 by 11 by hand and the remainders cycle 5, 6, 5, 6, which is what produces the repeating pair.
Worked example — a block that does not start immediately
Convert 0.1666…, where only the 6 recurs.
Shift once to move the non-recurring part across the point, then shift again by the block length.
The principle is unchanged: choose the two multipliers so the tails match, then subtract.
A common slip is to multiply by 10n where n is the number of decimal places shown rather than the length of the repeating block. For 0.454545… writing 1000000a is not wrong, but it is not helpful either unless the tails still align. Match the block length.
Real numbers and the numbers no fraction names
The rationals leave gaps. The diagonal of a unit square has length √2, which is a perfectly definite point on the line, and yet no fraction equals it. Numbers of this kind are irrational, and the reals are what you get by filling every such gap.
π
The ratio of a circle's circumference to its diameter. Irrational, and moreover not the root of any polynomial with integer coefficients.
√2
The diagonal of a unit square. Irrational, but it does satisfy x2 - 2 = 0, so it is an algebraic number.
3√7
The edge of a cube of volume 7. Irrational, and a root of x3 - 7 = 0.
0.454545…
Not irrational — the repeating block gives it away. It equals 5/11 exactly.
The source notes mark √2, π and similar values on the number line alongside the integers, which is the right picture: the irrationals are not exotic outliers, they are densely packed everywhere between the fractions.
Irrational values sit between the integers, not beyond them. √2 ≈ 1.414 and π ≈ 3.142, marked here as the source notes mark them.
Where the reals run out
One equation still has no solution. Because a real number squared is never negative, x2 + 9 = 0 is unsolvable in ℝ. The source notes put the two cases side by side, which is the clearest way to see what is at stake.
| Equation | Rearranged | Over the reals | Over the complex numbers |
|---|---|---|---|
| x2 - 9 = 0 | x2 = 9 | x = ±3 | x = ±3 |
| x2 + 9 = 0 | x2 = -9 | no solution | x = ±√-9 = ±3i |
A complex number is then anything of the form a + bi with a and b real. Taking b = 0 recovers the reals, which is why ℝ ⊂ ℂ. The arithmetic is developed in Complex Numbers: Arithmetic and the Conjugate.
It is worth being precise about the scope of the extension. Adding i does not just solve x2 = -9; it solves every polynomial equation with complex coefficients. No further extension is needed for algebra, which is a much stronger statement than the motivating example suggests.
Deciding which set a number belongs to
In practice the question is usually 'is this rational?', and the answer follows from the form the number arrives in.
- Is it a ratio of integers? If it can be written p/q with q ≠ 0, it is rational. This includes every integer, since 6 = 6/1.
- Is it a terminating decimal? Then it is rational: put the digits over the appropriate power of ten and reduce.
- Is it a recurring decimal? Then it is rational: use shift-and-subtract to produce the fraction.
- Is it a root that does not come out exactly? √n is irrational unless n is a perfect square; 3√n is irrational unless n is a perfect cube.
- Does it involve i with a non-zero coefficient? Then it is complex but not real.
| Number | Smallest set containing it | Reason |
|---|---|---|
| 7 | ℕ | A counting number |
| -7 | ℤ | Negative, so not in ℕ |
| -34 | ℚ | A ratio of integers, not an integer |
| 1.3 | ℚ | Terminating decimal = 13/10 |
| 0.4545… | ℚ | Recurring block = 5/11 |
| √9 | ℕ | Equals 3; the radical is a disguise |
| √2 | ℝ | Irrational: no fraction equals it |
| π | ℝ | Irrational |
| 4 - 5i | ℂ | Non-zero imaginary part |
√9 is a trap worth meeting once. The radical sign suggests irrationality, but √9 = 3 exactly. Always evaluate before classifying.
How this is used downstream
The number sets are not an abstract preliminary. They determine which answers are admissible, and a solution that is correct in one set may be meaningless in another.
The source notes state plainly that inequalities apply only in the real numbers. That restriction is easy to read past and important to keep: there is no way to order the complex numbers that respects the arithmetic.
Common mistakes
| Mistake | Why it happens | Check |
|---|---|---|
| Calling 0.333… irrational | The decimal never stops, which looks like irrationality | Look for a repeating block. There is one, so it is 1/3. |
| Writing pq without stating q ≠ 0 | The condition is usually invisible until it bites | Every time a denominator contains a variable, record the excluded value immediately. |
| Treating √-9 as -3 | Confusing √-9 with -√9 | (-3)2 = 9, not -9. The correct value is 3i. |
| Assuming every root is irrational | The radical sign is read as a warning label | Test for a perfect square or cube first. |
| Ordering complex numbers | Habit carried over from the reals | Ask whether i > 0 or i < 0; neither can hold consistently. |
Frequently asked questions
Is zero a counting number?
In the convention used here, no. The counting numbers start at 1. Some other conventions include 0 and write the set as ℕ0. Neither is wrong; what matters is stating which convention is in force, because results about divisibility and induction can change at the boundary.
Is every decimal that stops a fraction?
Yes. A decimal that terminates after n places equals the digit string over 10n, so 0.375 = 375/1000 = 3/8. Terminating decimals are a special case of recurring decimals with a recurring block of zeros.
Why is √2 not a fraction?
If √2 = p/q in lowest terms then p2 = 2q2, so p is even; writing p = 2k gives 2k2 = q2, so q is even too. That contradicts 'lowest terms'. The number exists on the line but no fraction names it, which is exactly what 'irrational' means.
Where do the complex numbers come from?
From the demand that every quadratic have a solution. x2 + 9 = 0 has none among the reals because a real square is never negative. Defining i by i2 = -1 supplies one, and it turns out to supply enough for every polynomial equation.
Source. Handwritten teaching notes, Week 1, pages 1-2, supplemented by the typed notes on fractions.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.