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You have used natural numbers since before you could read. But try to say exactly what they are, without just listing them, and it gets hard. In 1889 an Italian mathematician wrote down a short list of rules from which all of ordinary arithmetic can be built. Here is who Giuseppe Peano was, what his axioms say, two worked examples you can follow line by line, and what students can learn from his way of working.
Who was Giuseppe Peano?
The MacTutor biography of Peano, from the University of St Andrews, says he was born on 27 August 1858 in a farmhouse about 5 km from Cuneo in northern Italy, and as a boy walked the 5 km to school and back every day. He became a professor at Turin and died on 20 April 1932.
MacTutor lists a remarkable range: the axioms of 1889; in 1890, the first "space-filling" curve, a continuous curve that passes through every point of a square; a journal, Rivista di matematica, founded in 1891; and the Formulario Mathematico, a huge project from 1892 to 1908 to write all of mathematics in symbols. From 1903 he also worked on an international language, Latino sine flexione, Latin without grammar.
What did Peano publish in 1889?
The pamphlet was Arithmetices principia, nova methodo exposita (roughly, "the principles of arithmetic, set out by a new method"), printed in Turin by the Bocca brothers. A scan is free on the Internet Archive. In the preface Peano separates propositions proved by logic, which he calls theorems, from those that are not, which he calls axioms, and states that there are nine axioms. He also names a recent booklet by Richard Dedekind, Was sind und was sollen die Zahlen? (1888), as useful to him.
His section 1 begins by explaining a few undefined signs: N means a positive whole number, 1 means unity, a + 1 means the number following a, and = means equality. Then come the axioms. Translated from his symbols, the ones about numbers say:
1 is a number.
If a is a number, then a + 1 is a number.
If a and b are numbers, a = b exactly when a + 1 = b + 1.
If a is a number, a + 1 is not equal to 1.
Induction: if a set contains 1, and whenever it contains a number x it also contains x + 1, then it contains every number.
The remaining axioms in his list deal with properties of equality. Item 10 defines 2 = 1 + 1, 3 = 2 + 1, 4 = 3 + 1, and so on. Many modern books state a similar list starting from 0 instead of 1.
Worked example 1: proving 2 + 2 = 4
Peano then defines addition (his item 18): a + (b + 1) = (a + b) + 1. In words: adding "the next number after b" gives the number after a + b. The pamphlet itself shows a + 2 = a + (1 + 1) = (a + 1) + 1.
2 + 2 = 2 + (1 + 1), because 2 = 1 + 1.
= (2 + 1) + 1, by the definition of addition with a = 2, b = 1.
= 3 + 1, because 3 = 2 + 1.
= 4, because 4 = 3 + 1.
Every step quotes a definition. Nothing is taken on trust, not even something you "just know".
Worked example 2: induction in action
Claim: 1 + 2 + ... + n = n(n + 1)/2 for every natural number n.
Base case, n = 1. Left side 1. Right side 1 × 2 / 2 = 1. True.
Step. Assume it is true for n = k: 1 + ... + k = k(k + 1)/2.
Then 1 + ... + k + (k + 1) = k(k + 1)/2 + (k + 1) = (k + 1)(k/2 + 1) = (k + 1)(k + 2)/2.
That is the formula with n = k + 1. So the set of numbers for which the claim holds contains 1 and is closed under "add 1". By the induction axiom it contains every number.
Check with n = 4: 1 + 2 + 3 + 4 = 10, and 4 × 5 / 2 = 10.
What mistakes do students make with induction?
Skipping the base case. The step alone can "prove" false statements.
Assuming what you want to prove. You may assume the claim for k, not for k + 1.
Checking a few values and stopping. Examples are evidence, not proof.
Where does this sit in your syllabus?
The current NCERT Class 11 mathematics textbook (reprint 2026-27) has no chapter on mathematical induction in its contents, and the JEE (Main) 2026 syllabus does not list it. Induction is still standard in olympiad mathematics and in university courses, so it is worth learning if you are preparing for INMO; see our guide to proof writing for INMO. Check the syllabus for your own exam year.
If you want a guided path through proof and number ideas, the Fermi Concept Library covers every concept in mathematics with its own lesson.
Who did Peano influence?
Bertrand Russell's Nobel Prize biography says that at the 1900 mathematical congress in Paris he was impressed with the ability of Peano and his pupils, and immediately studied Peano's works. Russell's own discovery in set theory is told in Bertrand Russell and the paradox in set theory.
What can a student take from how Peano worked?
Write down your assumptions. Errors in proofs often hide in steps that "everyone knows".
Good notation is a tool. Peano's symbols, including a form of the ∈ sign for "is a member of", let him check steps mechanically.
Know your audience. MacTutor records that when he taught engineering students entirely from his symbolic Formulario, they found it above their heads. Precision needs clear explanation too.
Next step: prove by induction that 1 + 3 + 5 + ... + (2n − 1) = n², writing the base case and the step separately.
