# Number Systems

Mathematics I · Arithmetic and Number Theory · https://tryals.app/learn/mathematics-i/number-systems

## Building the Number Line

Mathematics builds numbers in stages to solve problems older systems could not. Each extension is driven by **closure**: enlarging a set so operations always yield an answer.

| Set | Symbol & Definition | Solves / Limits |
| --- | --- | --- |
| **Natural** | $\mathbb{N} = \{0, 1, 2, \dots\}$ | Counting. Fails at $3 - 5$. |
| **Integer** | $\mathbb{Z} = \{\dots, -1, 0, 1, \dots\}$ | Adds negatives. Fails at $3 \div 2$. |
| **Rational** | $\mathbb{Q} = \{\frac{a}{b} \mid b \neq 0\}$ | Fractions. Decimals terminate/repeat. |
| **Real** | $\mathbb{R} = \text{all decimals}$ | Adds irrationals ($\sqrt{2}, \pi$). No holes. |

These sets are **nested**: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. Every natural **is** an integer, and every integer **is** a rational (e.g., $5 = \frac{5}{1}$).

## Nested Systems & Pitfalls

Knowing which system a number lives in tells you which operations are guaranteed to stay inside it.

**Common pitfall:** Assuming every real number is rational, or that these systems are separate categories rather than a chain of ever-larger sets.

- **Irrationals** like $\sqrt{2}$ and $\pi$ are real but **not** rational; their decimal expansions neither terminate nor repeat.
- **Closure rules:** Naturals are closed under addition and multiplication, but not subtraction. Integers are closed under subtraction. Rationals are closed under division by nonzero values.

Each expansion unlocks roots, limits, and total operations. Always check your number system before assuming an operation has an internal solution.

## Practice questions

5 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. Select every statement that is TRUE.

A. Every rational number is an integer
B. Every integer is a rational number
C. Every natural number is a real number
D. The square root of 2 is rational
E. Every real number is a rational number
F. Some rational numbers are not integers

**Answer:** B. Every integer is a rational number; C. Every natural number is a real number; F. Some rational numbers are not integers

**Why:** Inclusion runs upward, never downward: 5 = 5/1 is rational, and every natural is real. But 1/2 refutes 'every rational is an integer', and the square root of 2 refutes 'every real is rational'. Reversing a nesting is the classic error here.

Page: https://tryals.app/practice/mathematics-i/number-systems/select-every-statement-that-is-true

### 2. Cantor showed the reals cannot be listed even in principle, while the rationals can. Both are infinite. What does the diagonal argument establish?

A. That infinity is not a well-defined mathematical notion
B. That the reals cannot be written down individually
C. That the reals are unbounded whereas the rationals are not
D. That infinities differ in size, since no list of reals is complete

**Answer:** D. That infinities differ in size, since no list of reals is complete

**Why:** Change the nth digit of the nth entry and you have a real that differs from every listed number somewhere, so the list was incomplete, whatever it was. The rationals survive the same challenge because they can be enumerated by a zig-zag, which is why 'infinite' turned out to name more than one size.

Page: https://tryals.app/practice/mathematics-i/number-systems/cantor-showed-the-reals-cannot-be-listed-even-in-principle-while-the

### 3. The real numbers were introduced to give $x^{2} = -1$ a solution.

**Answer:** False

**Why:** False. The reals fill the gaps the rationals leave, $x^{2} = 2$ has no rational solution but does have a real one. $x^{2} = -1$ has no *real* solution either; that is what forced the complex numbers. Each extension is driven by a specific equation, and getting which equation right is the point.

Page: https://tryals.app/practice/mathematics-i/number-systems/the-real-numbers-were-introduced-to-give-x-1-a-solution

### 4. Every fraction has a decimal expansion that either terminates or eventually repeats, and no irrational number does either. Why does that difference follow from what a fraction is?

A. Because fractions are simpler numbers than irrationals, so their decimal expansions are correspondingly simpler.
B. Because division by b has only b remainders, so one must recur.
C. Because terminating and repeating expansions are how fractions are defined, so the property holds by definition.
D. Because irrational numbers have infinitely many digits while rationals have finitely many.

**Answer:** B. Because division by b has only b remainders, so one must recur.

**Why:** Long division by b has only b possible remainders, so within b steps one must recur, and from there the digits repeat because the division is identical. A remainder of 0 is the terminating case. The converse completes it: a repeating expansion converts back to a fraction, so 0.101001000100001… is irrational by construction.

Page: https://tryals.app/practice/mathematics-i/number-systems/every-fraction-has-a-decimal-expansion-that-either-terminates-or

### 5. Each extension of the number system is driven by closure. Sort each expression by the smallest system in which it is guaranteed to have an answer.

**Answer:**

- Defined in the integers: the difference of two integers, the product of two integers
- Needs the rationals: 3 divided by 2, 1 divided by 7
- Needs the reals: the square root of 2, pi

**Why:** Each enlargement exists to make one operation total: negatives make subtraction always work, fractions make division by a nonzero number always work, and the reals supply the roots and limits that fractions cannot reach. Closure is the engine of the whole hierarchy.

Page: https://tryals.app/practice/mathematics-i/number-systems/each-extension-of-the-number-system-is-driven-by-closure-sort-each