"Theorems" Example Sentences
1. Euclid's five famous theorems are still studied in geometry today.
2. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
3. In calculus, the mean value theorem states that there is a point where the slope of the tangent line to a curve equals the average rate of change of the function over a given interval.
4. One of the most famous theorems in mathematics is Fermat's Last Theorem, which took over 350 years to prove.
5. In algebra, the fundamental theorem of algebra states that every polynomial of degree n has n roots, counting multiplicity.
6. The law of sines is a theorem that relates the angles and sides of a non-right triangle.
7. One of the most important theorems in number theory is the prime number theorem, which gives an estimate of the number of prime numbers below a given value.
8. The Cauchy-Schwarz inequality is a famous theorem in linear algebra and analysis.
9. In topology, the Jordan curve theorem states that a simple closed curve divides the plane into exactly two regions.
10. The Gauss-Bonnet theorem is a fundamental result in differential geometry that relates the curvature of a surface to its topology.
11. The Poincaré conjecture was a famous open problem in topology that was solved by Grigori Perelman in 2002, using advanced geometrical theorems.
12. In combinatorics, the Pigeonhole principle is a simple but powerful theorem that states that if you have n items and fewer than n containers, then one of the containers must hold at least two items.
13. One of the key theorems in probability theory is Bayes' theorem, which relates the conditional probabilities of events.
14. The Chinese remainder theorem is a theorem in number theory that tells us how to solve systems of congruences.
15. In set theory, Zorn's lemma is a theorem that is equivalent to the axiom of choice.
16. The Pythagorean triple theorem states that if a and b are positive integers such that a and b are relatively prime and one of them is even, then there exist integers x and y such that a=x^2-y^2, b=2xy, and c=x^2+y^2 form a Pythagorean triple.
17. Brouncker's theorem is a result in number theory that gives an infinite continued fraction representation of π.
18. The Riemann hypothesis is arguably the most famous unsolved problem in mathematics and concerns the distribution of prime numbers. It is related to many theorems in complex analysis and number theory.
19. In game theory, the Minimax theorem is a fundamental result that describes how players should choose their strategies in a one-on-one game with conflicting interests.
20. The Heine-Borel theorem is a result in analysis that states that a subset of Euclidean space is compact if and only if it is closed and bounded.
21. In linear algebra, the spectral theorem relates the eigenvalues of a symmetric matrix to its diagonalization.
22. The Banach-Tarski paradox is a counterintuitive result in measure theory that shows that it is possible to decompose a solid ball into a finite number of pieces, and reassemble them into two solid balls of the same size as the original.
23. The Knaster-Tarski fixed point theorem is an important result in topology and logic that characterizes the existence of fixed points of certain mappings.
24. The fundamental theorem of calculus connects differentiation and integration, showing that differentiation and integration of a function are inverse operations.
25. The Pythagorean theorem has many applications, such as in surveying, engineering, and physics.
26. In graph theory, the four color theorem states that any map on a plane can be colored with only four colors, such that no two adjacent regions have the same color.
27. The Hahn-Banach theorem is a fundamental result in functional analysis that gives a way to extend bounded linear functionals from a subspace to the whole space, while preserving certain properties.
28. The Lorenz gauge condition is a theorem in electromagnetism that specifies how the vector potential should be chosen to satisfy the equations of motion for electromagnetic waves.
29. The fundamental theorem of algebra is often used in solving polynomial equations, by guaranteeing that the equation has at least one complex root.
30. The Liouville theorem is a result in complex analysis that sets strict conditions on the possible behavior of entire functions.
Common Phases
1. "According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides;"
2. "The angle bisector theorem states that it divides the opposite side into segments proportional to the adjacent sides;"
3. "Fermat's little theorem states that if p is a prime number, then for any integer a, a to the p power minus a is an integer multiple of p;"
4. "The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root;"
5. "The mean value theorem states that for a differentiable function, there is at least one point within the interval where the instantaneous rate of change equals the average rate of change of the function."