Is 0 a natural number? It depends on the convention. Set theory, logic, computer science and the ISO standard say yes. Many school books in India, the US and Russia say no. Here are 10 memes and the full answer.
Is 0 a natural number? Ask ten mathematicians and you may get two loud camps and a few people who quietly leave the room. The argument has produced some of the best math memes on the internet, and we collected ten of them below. After the memes, you will find a clear explanation, a country-by-country guide and a look at what famous mathematicians chose.
1. Induction Starts at P(0)
This meme turns mathematical induction into a battle scene. Prove the base case P(0), then show P(n) leads to P(n+1), and the arrows cover every natural number.
2. Isn’t Zero Also a Natural Number?
The teacher writes {1, 2, 3, 4, 5…}. The student gasps and asks about zero. Every math class has had this exact moment at least once.
3. The Mathematician Afraid of Negative Numbers
He will “stop at nothing” to avoid negatives, so he stops at zero. The punchline, “that’s only natural,” lands on both meanings of the word.
4. The Natural Numbers IQ Bell Curve
Both ends of the curve use tally marks. The middle uses the von Neumann construction, where 0 is the empty set and each number is the set of all smaller ones.
5. Natural, Whole, Integers and Rational Numbers
This nested diagram follows the common school convention. Natural numbers start at 1, whole numbers add 0, integers add negatives, and rationals add fractions.
6. When Someone Asks If Zero Is a Natural Number
Some people treat this question like a hot political topic. The safest reply is to dodge it, and this meme shows exactly that move.
7. Mathematicians Debating Zero
Here the debate becomes a pirate ship standoff, with everyone pointing weapons at everyone else. It matches how heated the topic feels in a faculty lounge.
8. The Whole Number Wedding Disaster
A man says zero is not a whole number and loses his future father-in-law. Under the usual school rules, zero is whole but not natural.
9. The ℕ₀ Symbol
The small subscript zero removes all doubt. Writing ℕ₀ tells the reader that zero belongs to the set.
10. “Respect Other People’s Opinions” vs 0 ∉ ℕ
The tweet jokes that the “opinion” in question is 0 ∉ ℕ. Fans of zero treat that statement as a personal attack.
So, Is Zero a Natural Number?
The honest answer is yes and no, because mathematics has two valid conventions. In the first, the natural numbers are the counting numbers 1, 2, 3 and so on, so zero stays outside. In the second, the natural numbers are 0, 1, 2, 3 and so on, so zero is the first one.
The international standard ISO 80000-2 includes zero, and so do most set theorists, logicians and computer scientists. Many number theorists and school textbooks keep the older tradition and start at 1. You can read the full background in this detailed overview of the natural numbers, which also covers the notation.
Why Zero Fits the Natural Numbers
- Set theory: zero equals the size of the empty set, so counting “nothing” is a real count.
- Peano axioms: most modern versions begin with “0 is a natural number.” Read the Peano axioms to see how the chain of successors works.
- Algebra: with zero, addition has an identity element, so the set forms a clean monoid.
- Programming: arrays, loops and indexes start at 0, which feels natural to coders.
Why Some Books Leave Zero Out
- History: people counted sheep and days long before anyone wrote a symbol for “none.”
- Number theory: prime factorization and divisibility often work best with positive integers.
- School tradition: teachers keep a clear line between “natural” numbers and “whole” numbers.
A Short History of Zero and Counting
Early number systems had no symbol for zero. Medieval Easter calculators led by Dionysius Exiguus used zero as a number around the year 525, even without a Roman numeral for it. In 628 CE, the Indian mathematician Brahmagupta wrote rules for adding and subtracting with zero.
The big shift came in the 1880s. Dedekind and Peano built the natural numbers from axioms, and later set theorists started the chain at the empty set. That is why logic and computer science favor zero today. A short guide to natural numbers confirms that definitions still differ.
Is Zero a Natural Number in Each Country?
No single country decides this for the whole world. Each education system picks a convention, and exam boards publish their own notation lists. Here is the common pattern for school math.
| Country or standard | Is 0 a natural number? | Quick note |
|---|---|---|
| ISO 80000-2 (international) | Yes | ℕ starts at 0, and ℕ* marks the version without zero. |
| France | Yes | The Bourbaki tradition shapes French classes. ℕ includes 0 and ℕ* excludes it. |
| Italy | Usually yes | Most textbooks write ℕ with zero included. |
| Poland | Usually yes | Schools and universities commonly count 0 as natural. |
| Brazil | Yes | Textbooks list ℕ as 0, 1, 2, 3 and onward. |
| China | Yes | Modern national standards and school books include 0. |
| Germany | Mixed | Both ℕ and ℕ₀ appear, and many authors add the subscript to be clear. |
| United Kingdom | Mostly no in school | Primary classes use “counting numbers” from 1. Check your exam board’s notation list. |
| United States | No in school, mixed in college | K-12 standards start at 1 and call 0 a whole number. Computer science courses include 0. |
| India | No | NCERT books separate natural numbers (from 1) and whole numbers (from 0). |
| Russia and former USSR | No | The long tradition starts the natural numbers at 1. |
| Japan and South Korea | Usually no in school | School books start at 1, while some university texts include 0. |
Curricula change over time, so always check your own syllabus or textbook before an exam.
What Famous Mathematicians Chose
The split shows up in the history of mathematics too. Here is how some well-known names handled zero.
- Giuseppe Peano: his 1889 axioms started at 1. He later switched to a version that began with 0.
- Richard Dedekind: his 1888 axiom system began with 1.
- Edmund Landau: his famous analysis text built the natural numbers from 1.
- Bertrand Russell: he counted 0 among the natural numbers.
- John von Neumann: his construction defines 0 as the empty set, the same idea shown in the bell curve meme.
- Nicolas Bourbaki: the group included 0, which explains the French convention.
- Paul Halmos, Stephen Kleene and John Horton Conway: all treated 0 as a natural number.
- George A. Wentworth: this textbook author also included 0, long before computers made it fashionable.
Most early authors left zero out, so the trend toward including it is fairly modern.
Notation Cheat Sheet
- ℕ: the natural numbers, and the meaning depends on the author.
- ℕ₀: natural numbers with 0 included, so {0, 1, 2, 3…}.
- ℕ* or ℕ⁺: natural numbers without 0, so {1, 2, 3…}.
- ω: the set theory symbol for the set of natural numbers, starting at 0.
When you write a proof or an exam answer, state your convention in the first line. That one sentence prevents most arguments.
FAQ: Is 0 a Natural Number?
Is 0 a natural number or a whole number?
Zero is always a whole number under the school definition. Whether it is also a natural number depends on the convention you follow.
Is 0 a natural number in India?
No. Indian school books use natural numbers from 1 and call the set with 0 the whole numbers.
Is 0 a natural number in computer science?
Yes. Computer science almost always starts at 0, which matches indexes, counters and the set theory view.
Is zero a positive number?
No. Zero is neither positive nor negative in most countries, though French tradition calls it both.
What is the smallest natural number?
It is 1 in the school convention and 0 in the ISO and set theory convention.
Conclusion
So, is 0 a natural number? Yes in set theory, logic, computer science, France, Brazil, China and the ISO standard. No in many school systems, including India, Russia and the US. Neither side is wrong. Pick one convention, say it out loud, and enjoy the memes.
