Real number system — ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ: Question bank

Some words you must have straight before you start (each is built fully in the parent):

Closed under an operation :::: doing that operation on members of the set always lands you back inside the set.
Dense :::: between any two members you can always squeeze in another member.
Countable :::: you can line the set up as a first, second, third… list that eventually reaches every element.

True or false — justify

TF1. Every integer is a rational number.

True — any integer

n

equals

n /1

, a ratio of integers with nonzero denominator, so

Z \subset Q

TF2. Every rational is an integer.

False —

1/2

is a ratio of integers but is not a whole step on the number line, so

Q \neq = Z

TF3.

9

is irrational because it has a square root sign.

False —

9 = 3

, a perfect square simplifies to an integer; irrationality depends on the value, not the notation.

TF4. The sum of two irrational numbers is always irrational.

False —

2 + (- 2) = 0

, which is rational; irrationals are not closed under addition.

TF5. The product of a nonzero rational and an irrational is always irrational.

True — if

r \cdot x

were rational for rational

r \neq = 0

, then

x = (r \cdot x) / r

would be rational too, contradicting

x

irrational.

TF6.

0.999 \dots

is a number strictly less than

1

False — it equals exactly

1

; the repeating decimal

0. \overline{9}

is just another name for

1

(see Decimal Representations).

TF7.

Q

and

Z

have the same number of elements.

True — both are countably infinite (

ℵ_{0}

); a clever zig-zag list of

Q

pairs it one-to-one with

N

(see Cardinality and Infinity).

TF8. Because

Q

is dense, it has no gaps, so it fills the whole line.

False — density means no two members are "adjacent", but holes like

2

still remain; density

\neq =

completeness (see Completeness Axiom).

TF9.

π

is irrational, therefore

π

is not a real number.

False — irrational numbers are real;

R = Q \cup (irrationals)

, so

π \in R

TF10. Every decimal that never terminates is irrational.

False —

0.333 \dots = 1/3

never terminates but repeats, and repeating decimals are rational; only non-repeating, non-terminating decimals are irrational.

Spot the error

SE1. "

3 - 5 = - 2 \in / N

, so subtraction is broken." — What's wrong?

Nothing is broken;

N

simply isn't closed under subtraction. That gap is exactly the motivation for inventing

Z

SE2. "In the

2

proof we assume

2 = p / q

; that assumption already uses

2

, so it's circular." — What's wrong?

It's proof by contradiction: we suppose it's rational to derive an impossibility. Reaching a contradiction refutes the supposition, which is valid, not circular.

SE3. "

p^{2}

is even, so

p

could be even or the square could just happen to be even." — What's wrong?

p

were odd,

p^{2} = 2 (2 k^{2} + 2 k) + 1

is odd; so an even

p^{2}

forces

p

even. There is no "just happens" case.

SE4. "

N \subseteq Z

and

N \subset Z

mean the same thing, so use either." — What's wrong?

\subseteq

allows equality;

\subset

(proper) additionally demands

Z

has an element like

- 3

not in

N

. Here the proper form is the accurate one.

SE5. "Between

a

and

b

take

(a + b) /2

; since we did it once, that's the only rational between them." — What's wrong?

You can repeat the averaging forever on the new gaps, producing infinitely many rationals between any two. One construction does not imply uniqueness.

SE6. "

R

is infinite and

N

is infinite, so they're the same size." — What's wrong?

Infinities come in sizes. Cantor's diagonal argument shows

R

can't be listed, so

∣ R ∣ = 2^{ℵ_{0}} > ℵ_{0} = ∣ N ∣

(see Cantor's Diagonal Argument).

SE7. "

(5, 2)

and

(6, 3)

are different integers because the pairs differ." — What's wrong?

The pair

(a, b)

stands for

a - b

; both give

3

, so they represent the same integer. Equality of integers is about the difference, not the raw pair.

Why questions

WHY1. Why is

Z

called a "ring" but

Q

a "field"?

Both allow

+, -, \times

, but only

Q

also lets you divide by any nonzero element; that extra reciprocal is what upgrades a ring to a field (see Field Axioms).

WHY2. Why did we need to invent irrationals if rationals are already dense?

A right triangle with legs

1

forces a diagonal of length

2

, which is not any

p / q

; without irrationals such genuine lengths would have no number, leaving real gaps.

WHY3. Why does the density proof require dividing by

2

rather than some other step?

Averaging

(a + b) /2

is guaranteed to sit strictly between

a

and

b

and stays rational (sum and division of rationals is rational), so it always lands a valid new member in the gap.

WHY4. Why can't "

\infty

" be the largest natural number?

\infty

is not a natural number at all; for any

n \in N

you can form

n + 1 \in N

, so no member is largest — there is no top to reach.

WHY5. Why is

Q

countable even though it looks "denser" than

Z

Visual denseness has nothing to do with cardinality; a zig-zag enumeration of

Z \times Z

still hits every fraction in a single list, matching them to

N

WHY6. Why do we write

∣ R ∣ = 2^{ℵ_{0}}

instead of

ℵ_{1}

Whether

2^{ℵ_{0}} = ℵ_{1}

is the Continuum Hypothesis, which is independent of the standard axioms, so we only assert the safe form (see Continuum Hypothesis).

WHY7. Why is

π

irrational but

2

irrational for a different kind of reason?

2

is algebraic (a root of

x^{2} - 2 = 0

) yet irrational;

π

is transcendental — root of no polynomial with integer coefficients — a strictly stronger property (see Algebraic vs. Transcendental Numbers).

Edge cases

EC1. Is

0

a natural number?

It depends on convention —

N = {1, 2, \dots}

excludes it while

N_{0} = {0, 1, 2, \dots}

includes it; always state which you mean. It is unambiguously an integer.

EC2. Is

0

rational?

Yes —

0 = 0/1

is a ratio of integers with nonzero denominator, so

0 \in Q

EC3. Is

p /0

a rational number?

No — the definition demands

q \neq = 0

; division by zero is undefined, so no such rational exists.

EC4. Is the set

{- 1, 0, 1}

closed under multiplication? Under addition?

Closed under multiplication (every product lands in the set) but not under addition, since

1 + 1 = 2 \in / {- 1, 0, 1}

(see Properties of Operations).

EC5. Is

2 \cdot 2

irrational?

No — it equals

2

, a rational; two irrationals can multiply to a rational, so irrationals aren't closed under multiplication either.

EC6. Does a terminating decimal like

0.5

count as rational?

Yes —

0.5 = 1/2

; every terminating decimal is a fraction with a power-of-ten denominator, hence rational.

EC7. Is the empty gap between

\frac{1}{3}

and

\frac{1}{2}

ever truly "next to" a rational?

No — no rational has a nearest neighbour; density guarantees another rational always fits between, so "adjacent rationals" never exist.

Recall Quick self-check

Countable vs uncountable — which is $R$ ? ::: Uncountable, with $∣ R ∣ = 2^{ℵ_{0}} > ℵ_{0}$ . Density vs completeness — which does $Q$ have? ::: Density only; completeness (no gaps at all) belongs to $R$ . Return to the parent note to see these ideas in their original setting.