Extended real numbers

the real numbers with ∞ and -∞

If you take the real numbers and add $+\infty$ and $-\infty$, you get what’s called the extended real number line (often written as $\overline{\mathbb{R}}$):

$$ \overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}. $$

1. What it is

• $\mathbb{R}$ is the set of all real numbers: all the decimals, fractions, irrationals, negatives, zero, and positives.
• Adding $+\infty$ and $-\infty$ gives you a closed, bounded line in the order sense — you can think of it as the number line with endpoints.

2. Why it’s used

The extended real line is especially handy in:

• Limits: You can write $\lim_{x \to 0^+} \frac{1}{x} = +\infty$ without needing to treat $+\infty$ as a “real” number in the usual sense.
• Optimization: In convex analysis, functions can take the value $+\infty$ to represent “not allowed” or “infeasible.”
• Measure theory & integration: Infinite integrals are expressed neatly when $+\infty$ is an allowable value.

3. Properties

• Order: $-\infty < x < +\infty$ for all $x \in \mathbb{R}$.
• Arithmetic quirks: Some operations work nicely (e.g., $x + (+\infty) = +\infty$ for any finite $x$), but others are undefined (like $\infty - \infty$).
• Topology: It’s like the real line but compact in the order topology; sequences can “converge” to $+\infty$ or $-\infty$.

4. Related ideas

• Projective real line: Instead of two infinities, it has just one ($\infty$), “wrapping” the line into a circle.
• Riemann sphere: In complex analysis, $\mathbb{C}$ gets one extra infinity point.

If you want, I can show you a diagram of $\overline{\mathbb{R}}$ and how it compares to the projective real line.

Algebraic properties

Alright — if we treat the real numbers together with $+\infty$ and $-\infty$ as the extended real line $\overline{\mathbb{R}}$, the algebraic rules change because $+\infty$ and $-\infty$ aren’t ordinary numbers. They’re symbols with special rules that mostly come from limit behavior.

Here’s a careful breakdown.

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1. Addition

For finite $x \in \mathbb{R}$:

$$ x + (+\infty) = +\infty, \quad x + (-\infty) = -\infty $$

Also:

$$ (+\infty) + (+\infty) = +\infty, \quad (-\infty) + (-\infty) = -\infty $$

Undefined case:

$$ (+\infty) + (-\infty) \quad \text{is undefined.} $$

It could arise from limits going different ways, so it’s left undefined.

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2. Multiplication

For finite $x > 0$:

$$ x \cdot (+\infty) = +\infty, \quad x \cdot (-\infty) = -\infty $$

For finite $x < 0$:

$$ x \cdot (+\infty) = -\infty, \quad x \cdot (-\infty) = +\infty $$

Also:

$$ (+\infty) \cdot (+\infty) = +\infty, \quad (-\infty) \cdot (-\infty) = +\infty $$

$$ (+\infty) \cdot (-\infty) = -\infty $$

Undefined cases:

$$ 0 \cdot (+\infty), \quad 0 \cdot (-\infty) \quad \text{are undefined.} $$

These show up as “indeterminate forms” in calculus.

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3. Division

For finite $x > 0$:

$$ \frac{x}{0^+} = +\infty, \quad \frac{x}{0^-} = -\infty $$

For finite $x < 0$:

$$ \frac{x}{0^+} = -\infty, \quad \frac{x}{0^-} = +\infty $$

Also:

$$ \frac{\text{finite nonzero}}{+\infty} = 0, \quad \frac{\text{finite nonzero}}{-\infty} = 0 $$

Undefined cases:

• $\frac{\infty}{\infty}$, $\frac{-\infty}{\infty}$, $\frac{\infty}{-\infty}$ — all are indeterminate.
• $\frac{0}{0}$ — indeterminate.
• $\frac{\infty}{0}$ — not defined in the usual extended real number system (though in limits, it might be interpreted as $\infty$).

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4. Negation

$$ -(+\infty) = -\infty, \quad -(-\infty) = +\infty $$

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5. Order properties

• Still totally ordered:

$$ -\infty < x < +\infty \quad \text{for all } x \in \mathbb{R}. $$

• Min/max are well-defined: $\min(x, +\infty) = x$, $\max(x, -\infty) = x$ for finite $x$.

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6. Field structure?

This is not a field:

• No additive inverses for $+\infty$ and $-\infty$.
• Not closed under addition or multiplication in the usual sense (because of undefined forms).
• Best to think of it as an ordered set with partial algebra designed to extend limits.

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If you’d like, I can give you a full algebraic operations table for $\overline{\mathbb{R}}$ so you can see all the defined and undefined cases at a glance. That would make the rules very clear.