# Real Numbers and the Number Line

Mathematics I · Arithmetic and Number Theory · https://tryals.app/learn/mathematics-i/real-numbers-and-the-number-line

## The Continuous Line

The **real numbers** $\mathbb{R}$ fill every gap of the rationals, forming a continuous line. Their defining trait is **completeness**: every set bounded above has a **least upper bound** (supremum) in $\mathbb{R}$. The rationals fail this, as the set of rationals with square under 2 misses $\sqrt{2}$.

Positions and distances are measured via tools:

| Tool | Notation / Form | Meaning |
| --- | --- | --- |
| **Closed Interval** | $[a, b]$ | Endpoints included |
| **Open Interval** | $(a, b)$ | Endpoints excluded |
| **Absolute Value** | $|x|$ | Distance to 0 ($x$ if $\ge 0$, else $-x$) |
| **Distance** | $|x - y|$ | Gap between $x$ and $y$ |

## Inequalities and Bounds

A set is **bounded above** if a number exceeds all its elements (an **upper bound**). **Completeness** ensures the supremum exists in $\mathbb{R}$, though it need not belong to the set itself.

The **triangle inequality** $|x + y| \le |x| + |y|$ states the combined step length never exceeds the sum of parts. It is vital for analysis.

> **Common pitfall:** Confusing _an_ upper bound with the _least_ upper bound. Any number above the set works, but the supremum is the single smallest one.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Compute the value of |3 - 11| + |-4|.

**Answer:** 12

**Why:** |3 - 11| = |-8| = 8 and |-4| = 4, so the total is 12. Note that |3 - 11| is also the distance between 3 and 11 on the line, the reading of |x - y| that makes absolute value genuinely useful.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/compute-the-value-of-3-11-4

### 2. Sort each number by whether it lies in the interval from -2 to 3, including -2 but excluding 3.

**Answer:**

- In the interval: -2, 0, 2.999
- Not in the interval: 3, -2.5, 3.0001

**Why:** The interval is half-open: -2 belongs but 3 does not, however tiny the gap. 2.999 is inside, 3.0001 is outside, and -2.5 falls below the left endpoint. Closeness is irrelevant, only the strict inequality decides membership.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/sort-each-number-by-whether-it-lies-in-the-interval-from-2-to-3

### 3. Complete the completeness property of the real numbers.

**Answer:** Every non-empty set of reals that is bounded above has **a least upper bound that is itself a real number**.

**Why:** Completeness guarantees a least upper bound, not a maximum, the set of x below 5 has a supremum but no largest element. The rationals fail this: the rationals whose square is below 2 are bounded above but their least upper bound, the square root of 2, is missing from the rationals.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/complete-the-completeness-property-of-the-real-numbers

### 4. Let S be the set of all rational numbers q with q squared less than 2. Select every TRUE statement.

A. S has a largest element
B. S has a least upper bound that is a rational number
C. S is bounded above
D. S has a least upper bound in the real numbers
E. The supremum of S is the square root of 2

**Answer:** C. S is bounded above; D. S has a least upper bound in the real numbers; E. The supremum of S is the square root of 2

**Why:** S is bounded above (by 2, say) and its supremum in the reals is the square root of 2, but that is irrational, so no RATIONAL least upper bound exists. This missing boundary is precisely the gap in the rationals that completeness fills. S also has no largest element, since you can always squeeze in a closer rational.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/let-s-be-the-set-of-all-rational-numbers-q-with-q-squared-less-than

### 5. Completeness is what separates ℝ from ℚ, and almost every theorem of calculus depends on it. Which of these fails over the rationals?

A. The definition of a derivative as a limit of difference quotients
B. The rule that a sum of continuous functions is continuous
C. The claim that polynomials are continuous everywhere
D. The intermediate value theorem, with no rational root between 1 and 2

**Answer:** D. The intermediate value theorem, with no rational root between 1 and 2

**Why:** The function changes sign and never hits zero, because the crossing point is missing from ℚ. That is completeness doing invisible work, the same property underwrites the extreme value theorem, convergence of bounded monotone sequences and the existence of maxima, which is why calculus is built on ℝ.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/completeness-is-what-separates-from-and-almost-every-theorem-of

### 6. How many integers x satisfy |x - 4| < 3?

**Answer:** 5

**Why:** |x - 4| < 3 means 1 < x < 7, so the integers are 2, 3, 4, 5, 6, five of them. The endpoints 1 and 7 are excluded because the inequality is strict; including them would give 7, which is the standard off-by-two error.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/how-many-integers-x-satisfy-x-4-3

### 7. When examining why the rational number line fails completeness, why does the bounded collection of fractions with squares below 2 expose an unfilled gap?

A. Because the set is infinite, and infinite sets of rationals never have least upper bounds.
B. Since the supremum is √2, which is an irrational number.
C. Because the rationals are not ordered densely enough to support least upper bounds.
D. Because the set has no upper bound at all within the rationals, so no least one can exist.

**Answer:** B. Since the supremum is √2, which is an irrational number.

**Why:** Upper bounds are abundant — 1.5, 1.42, 1.415 — and there is no smallest one, because any rational bound can be lowered slightly and still clear the set. Density is not completeness: density says there are no jumps, completeness says no holes where a limit should be. Without it the intermediate value theorem and bounded monotone convergence both fail.

Page: https://tryals.app/practice/mathematics-i/real-numbers-and-the-number-line/when-examining-why-the-rational-number-line-fails-completeness-why