Infinity is one of those math concepts that stays with you. The moment you really think about it, it raises questions that feel almost impossible to answer.
What happens when you add to something that's already endless? Can you subtract infinity from itself? Is there more than one kind of infinity?
These are just a couple of questions we get from our students. We put together clear, honest answers to the ones that surface most often, building from the basics toward the truly mind-bending.
Here are the answers to the questions students ask most, starting from the basics and building toward the truly mind-bending.
Infinity is not a number but a concept, the idea of something without end. It describes a sequence of numbers that goes on forever, a line that never stops, or a set that keeps growing no matter how long you count.
In math, we denote infinity with the symbol ∞, introduced by English mathematician John Wallis in 1657. More on the symbol in question 7.
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Some mathematical situations simply can't be described with ordinary numbers. Without infinity, we couldn't precisely describe a line that stretches in both directions or explain what happens to a function as it grows without bound.
Infinity gives mathematicians a rigorous way to work with things that never end, and as you'll see in the questions below, it behaves in some truly surprising ways.
No, and this is one of the most important things to understand about infinity. Infinity is:
Not a natural number, whole number, or real number
Not something you can place on a standard number line
Not reachable by counting
Not usable in calculations the way ordinary numbers are
Infinity describes endlessness, and that distinction is exactly what makes all the operation questions below so interesting.
It is neither. Even and odd are properties of integers, whole numbers like 2, 7, or 44. Since infinity isn't an integer, the question simply doesn't apply.
It's a bit like asking whether the color blue is heavy: the words are real, but the combination doesn't make sense. Infinity lives outside the number system where even and odd exist.
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Mathematicians have wrestled with infinity for thousands of years. Ancient Greek philosophers like Zeno of Elea were raising paradoxes about infinite sequences as far back as the 5th century BC.
Around the same time, Jain mathematicians in ancient India were independently developing their own framework for infinity, even distinguishing between different types of infinite quantities, centuries before Cantor formalized the idea in the West.
But the person who transformed infinity into a serious branch of mathematics was Georg Cantor (1845–1918). He showed that infinity could be studied rigorously and made the astonishing discovery that some infinities are actually bigger than others. (More on that in the last question.)
The infinity symbol ∞ was introduced by English mathematician John Wallis in 1657. He chose the looping shape to represent endlessness in mathematical notation, and it's been in use ever since.
Fun fact: Even the most familiar symbols have an inventor and a birthday.
The equals sign (=) was introduced by Robert Recorde in 1557
The plus and minus signs (+ and -) appeared in print in the late 1400s
The percent symbol (%) evolved gradually over centuries from a handwritten abbreviation
The symbol ∞ is called a lemniscate, from the Latin word lemniscus, meaning “ribbon.” It looks like a figure eight lying on its side. He chose the looping, figure-eight shape to represent endlessness, and it has been in use ever since. You will spot it in:
Limits, as x gets larger and larger without bound
Calculus, to describe values that grow forever
Set theory, to compare the sizes of endless collections
Technically, it represents potential infinity, the idea of something that could go on forever rather than an actually infinite quantity, which is a more involved mathematical idea.
Here's the classic playground argument, and the answer surprises most students:
Infinity plus one is still infinity.
Because infinity isn't a fixed number, adding one to it doesn't make it larger, as it's already endless.
Think of it this way: if you have an infinite list of numbers and add one more item, the list is still infinite. In notation: ∞ + 1 = ∞.
Same idea, same result:
Infinity plus infinity is still infinity.
Combining two endless things doesn't produce something more endless. Endlessness plus endlessness is just endlessness. In notation: ∞ + ∞ = ∞.
This feels wrong at first because our arithmetic instincts are built for finite numbers, where adding always produces something larger. Infinity plays by different rules.
This one is truly different, and here's where things get really interesting.
Infinity minus infinity is not infinity. It's not zero either.
It's what mathematicians call an indeterminate form, meaning the answer can't be determined without more context. Depending on how the two infinities were produced, the result could be:
Zero
A specific number
Infinity itself
This is one of infinity's most surprising properties: ordinary subtraction rules simply don't apply.
Zero times infinity is another indeterminate form. Most students expect the answer to be zero, since anything times zero equals zero in ordinary arithmetic, but that rule doesn't apply to infinity.
The result depends entirely on context. This is where it gets interesting. Infinity isn't a fixed value; it's a moving target. Depending on how the two interact, the result could be:
Zero
A specific finite number
Infinity itself
This combination comes up in calculus, where specific techniques are used to work out what indeterminate forms actually equal in a given problem.
Also indeterminate. Students often expect the answer to be one, since any number divided by itself equals one, but because infinity isn't a fixed quantity, that rule breaks down, too.
Like ∞ − ∞ and 0 × ∞, this is a case where ordinary arithmetic simply can't be applied directly.
In standard arithmetic, dividing by zero is undefined. Calculators return an error because no valid answer exists.
But here is something worth noticing. As we divide a number by values closer and closer to zero, the result shoots toward infinity:
1 ÷ 0.1 = 10
1 ÷ 0.01 = 100
1 ÷ 0.001 = 1,000
The closer the divisor gets to zero, the bigger the result grows, all the way toward infinity. So while dividing by zero does not equal infinity, the two ideas are clearly linked.
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Yes. This might be the most mind-bending fact in all of mathematics.
Georg Cantor proved that not all infinities are the same size. Here's the key distinction:
The infinity of counting numbers (1, 2, 3, 4…) is one kind of infinity
The infinity of all real numbers (every decimal, fraction, and irrational number) is a provably larger kind of infinity
There are infinitely many real numbers between 0 and 1 alone, and that infinity is bigger than the infinity of all whole numbers
Mindboggling, right?!
This discovery was revolutionary. It changed the course of mathematics and challenged everything mathematicians thought they understood about size and quantity.
Today, Cantor's work underpins some of the most important areas of modern math, from calculus and mathematical analysis to computer science and physics, anywhere we need to reason carefully about things that go on forever.
At Mathnasium, curiosity isn't a distraction. It's the whole point.
Mathnasium is a math-only learning center that helps K–12 students catch up, keep up, and get ahead in math.
The curiosity that drives students to ask why infinity plus one still equals infinity, or how one infinity can be bigger than another, is exactly the kind of thinking we love to see.
At Mathnasium, when a student wonders about infinity, we take that seriously. Our seasoned tutors use those moments to deepen understanding, connecting big ideas to the foundational concepts students are already building in their personalized learning plans.
The best math students aren't just the ones who follow procedures. They're the ones who notice patterns, ask questions, and wonder why things work the way they do. That curiosity is what the Mathnasium Method™ is built to develop.
Here’s how we support lasting mastery:
Diagnostic assessment first: Each student starts here, so tutors know exactly where they are, where their knowledge gaps are, and what they're ready to explore next.
Personalized learning plans: Built from assessment insights, so each student moves forward at a pace that builds true understanding.
Specially trained tutors: Encourage questions and teach for true understanding, not memorization.
A caring and fun environment: Asking "but why does that work?" isn't a distraction. It's the whole point.
We work with students across the full K–12 range. The results speak for themselves:
94% of parents report an improvement in their child's math skills and understanding
93% of parents report their child's improved attitude toward math after attending Mathnasium
90% of students saw an improvement in their school grades
We operate over 1,100 learning centers across North America, bringing our top-rated method close to your community.
If you are based in or near Meridian, ID, Mathnasium of Meridian is a trusted local center with years of experience helping students excel in math.
Whether your child is looking to catch up, keep up, or get ahead, our team is happy to assist!
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