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- All abelian groups are solvable.
- The only factorial that's also a square is 1.
- The semi-direct product of two abelian groups may be non-abelian.
- The diagonal elements of a complex Hermitian matrix are all real.
- Every group of prime order is cyclic.
- Every subgroup of a cyclic group is cyclic.
- In a commutative ring with identity, every maximal ideal is prime.
- The final digits of the Fibonacci sequence have period 60.
- Representations of compact topological groups are semi simple.
- Every Polish space is homeomorphic to a subspace of the Hilbert cube.
- A holomorphic function of two or more complex variables has no isolated zeros.
- If p is prime, every group of order p² is abelian.
- Fermat's last theorem does not apply to integer matrices of size > 1.
- The characteristic of a field is either 0 or a prime.
- The derivative of a Bézier curve is another Bézier curve.
- In linear logic, multiplicative operators can be understood as context-free and additive operators as contextual.
- A monad is just a monoid in the category of endofunctors.
- In the category of Abelian groups, the direct sum is both a product and a coproduct.
- Laplace transforms turn differential equations into algebraic equations.
- The set of all algebraic numbers is countable.
- The set of all transcendental numbers is uncountable.
- If X is a set of Borel measure zero, not every subset of X must be Borel measurable. But every subset of X is Lebesgue measurable.
- A projective plane has Euler characteristic 1.
- The Euler characteristics of a torus and a Klein bottle are both 0.
- There are 8 semi-regular (Archimedean) tessellations.
- A space is simply connected if its fundamental group is trivial.
- A space is connected if it has no proper subsets that are both open and closed.
- All abelian simple groups are cyclic groups of prime order.
- Convex polygons of more than 6 sides cannot tessellate.
- The uniform limit of continuous functions is continuous.
- The fundamental group of the product of two spaces is the direct product of their fundamental groups.
- In Euclidean spaces, an open connected subset of the complex plane is path connected.
- The fundamental group of a projective plane P² has order 2.
- The product of regular spaces is regular.
- The product of Hausdorff spaces is Hausdorff.
- The product of compact spaces is compact.
- A metric space X is complete if every Cauchy sequence in X converges to a point in X.
- The product of path connected spaces is path connected.
- A space is connected if it is not the union of disjoint open sets.
- A field is a commutative division ring. In other words, it is an object in the category of rings such that every epimorphism from it is either an isomorphism or a morphism to a terminal object.
- A category is the horizontal categorification of a monoid.
- In a category with biproducts, there is a decomposition of any morphism in terms of arrays of morphisms between each object's summands. This array is called a matrix.
- The axiom of choice states that every non-empty set admits a group structure.
- A Lie group is a manifold and a group.
- An n x n matrix with n distinct eigenvalues is diagonalizable.
- The determinant of a matrix is the product of its eigenvalues.
- A group is called simple if it has no normal subgroups other than the identity and the group itself.
- All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic.
- It is unknown whether odd perfect numbers exist.
- The sum of the reciprocals of the primes diverges.
- Every positive integer can be written as a sum of distinct Fibonacci numbers.
- Every permutation can be written as a product of disjoint cycles.
- In a group G, the order of any element is a divisor of the order of G.
- If matrices A and B commute, then they have a common eigenvector.
- For each prime p and positive integer n, there is a unique field of order pⁿ.
- All finite division rings are commutative and therefore finite fields.
- Every field is an integral domain. Every finite integral domain is a field.
- The sum of Hermitian matrices is Hermitian.
- If p and q are prime, every group of order pᵐ qⁿ is solvable.
- There are no non-abelian simple groups of order less than 60.
- Any two finite fields having exactly pⁿ elements for prime p are isomorphic.
- In the category of Abelian groups, coproducts are direct sums.
- In the category of groups, coproducts are free products.
- Every positive integer is the sum of four squares. If n is congruent to 7 mod 8 then n is not a sum of three squares.
- The only number > 1 that is square and pyramidal is 4900.
- An algebraically closed field must be infinite.
- The order of a finite field must be a prime power.
- If a polynomial has real coefficients, complex roots come in conjugate pairs.
- The operation of putting a matrix into Jordan canonical form is discontinuous.
- Every ring homomorphism from a field to a non-zero ring is injective.
- The Riemann zeta function at positive even integers takes on values that are rational multiples of even powers of pi.
- A space is sequentially compact if every sequence has a convergent subsequence.
- Finite products of locally compact spaces are locally compact.
- The image of a compact set under a continuous function is compact.
- A compact subset of a Hausdorff space is closed.
- A compact Hausdorff space is normal.
- A closed subgroup of a Lie group is a submanifold.
- A closed subset of a compact space is compact.
- A function between two topological spaces is continuous if the inverse image of every open set is open.
- A closed subset of a normal space is normal.
- The field of p-adic numbers has a totally disconnected topology.
- A space is second countable if it has a countable basis.
- A space is separable if it contains a countable dense subset.
- A Hausdorff space is normal if every disjoint pair of closed sets can be separated by disjoint open sets.
- A complex elliptic curve is topologically a torus.
- An elliptic curve over a finite field is either a cyclic group or the product of two cyclic groups.
- Every non-reflexive Banach space contains two disjoint closed bounded convex sets which cannot be separated by a closed hyperplane.
- In calculus over the p-adic numbers, a function can have derivative zero without being constant.
- Weierstrass elliptic functions have one singularity of order 2 inside the fundamental parallelogram.
- A holomorphic function of two or more complex variables has no isolated zeros.
- A functor is called exact if it preserves short exact sequences.
- A functor is called faithful if it is injective on hom-sets.
- Every commutative monoid can be extended to a group.
- A ring R is Noetherian if every ideal of R is finitely generated.
- In the category of rings, an epimorphism is not necessarily surjective. Unlike sets and groups.
- Every finite abelian group is the direct sum of its nontrivial Sylow subgroups.
- A group G is simple if its only normal subgroups are the identity and G itself.
- The Krull dimension of a commutative ring R is the length of the longest chain of prime ideals in R.
- Every element in a finite field can be written as the sum of at most two squares.
- A group is solvable if every factor in its composition series is cyclic.
- An algebraic number field is a finite degree extension of the rationals.
- The nth prime is greater than n log n.
- The only Fibonacci numbers that are cubes are 1 and 8.
- Every odd integer n ≥ 7 is the sum of three primes.
- Every invertible complex matrix is the exponential of some complex matrix.
- Every finite field is a splitting field of a polynomial.
- A local ring is a ring that has exactly one maximal ideal.
- An elliptic curve can have only a finite number of points with integer coordinates.
- Every Boolean algebra is isomorphic to a field of sets.
- The ideals of a ring form a semiring.
- A wheel is an algebraic structure that is the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.
- For elliptic curves, it's easier to prove that addition is commutative than to prove it is associative.
- A real matrix has orthogonal eigenvectors if and only if it commutes with its transpose.
- The word “hundred” comes from the old Norse term, “hundrath”, which actually means 120 and not 100.
- The symbol for division (÷) is called an obelus.
- The symbol for infinity (∞) is called a lemniscate.
- A ring is a monoid internal to (Ab, ⊗, ℤ).
- A group is a groupoid with a single object.
- A groupoid is a category in which every morphism is invertible (an isomorphism).
- A category is called small if the collections of its objects and morphisms are sets.
- A monoid in a monoidal category is the categorical generalisation of a monoid.
- The sum of any two odd numbers or any two even numbers is an even number.
- 0.999... is exactly equal to 1. Not approximately equal, instead, they represent the very same number.
- An order preserving map on a complete lattice has a fixed point.
- A matrix is invertible if and only if its determinant is non-zero.
- The sum of the first n odd numbers is n².
- In topology, a coffee mug and a donut are homotopically equivalent.
- A group is called simple if it has no normal subgroups other than the trivial group and the group itself.
- The number of subsets of a set with n elements is 2ⁿ.
- In an infinite set, the number of elements and the number of subsets of those elements are incomparable.
- In a finite group, the number of elements that are their own inverses is always even.
- A number is divisible by 9 if and only if the sum of its digits is divisible by 9.
- A cyclic group of prime order has only the trivial group as a proper subgroup.
- The j-invariant of i is 1728.
- A limit is the terminal object in the category of cones of some functor.
- Jacques Tits is a Belgian mathematician, known for Tits buildings, the Tits alternative, the Tits group and the Tits metric.
- All tessellation types for convex polygons are known, except for pentagons.
- A linear operator is continuous if and only if it maps the unit ball to a bounded set.
- Every subgroup of a free group is free.
- Every integer greater than 77 is the sum of integers whose reciprocals sum to 1.
- There are infinitely many primes p such that p+2 is either prime or the product of two primes.
- The Mandelbrot set is connected.
- All separable Banach spaces are isometrically isomorphic to closed subspaces of C[0, 1].
- Laplace transforms turn differential equations into algebraic equations.
- The derivative of an odd function is an even function.
- A matrix and its transpose have the same rank.
- A midpoint convex function is not necessarily convex, but a strictly midpoint convex function is strictly convex.
- A set is convex iff its indicator function is a convex function. (Indicator function is infinite off the set).
- Every C¹ differential structure on a manifold can be uniquely smoothed to a C^∞ differential structure.
- The Hilbert transform is an anti-involution, i.e. H(H(u)) = -u.
- Jacobi elliptic functions have two singularities of order 1 inside the fundamental parallelogram.