The Mathematics Portal

Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

Refresh with new selections below (purge)

Featured articles - load new batch

Cscr-featured.png
  Featured articles are displayed here, which represent some of the best content on English Wikipedia.

Selected image – show another

animation of dots of varying heights being sorted by height using the quicksort algorithm
Quicksort (also known as the partition-exchange sort) is an efficient sorting algorithm that works for items of any type for which a total order (i.e., "≤") relation is defined. This animation shows how the algorithm partitions the input array (here a random permutation of the numbers 1 through 33) into two smaller arrays based on a selected pivot element (bar marked in red, here always chosen to be the last element in the array under consideration), by swapping elements between the two sub-arrays so that those in the first (on the left) end up all smaller than the pivot element's value (horizontal blue line) and those in the second (on the right) all larger. The pivot element is then moved to a position between the two sub-arrays; at this point, the pivot element is in its final position and will never be moved again. The algorithm then proceeds to recursively apply the same procedure to each of the smaller arrays, partitioning and rearranging the elements until there are no sub-arrays longer than one element left to process. (As can be seen in the animation, the algorithm actually sorts all left-hand sub-arrays first, and then starts to process the right-hand sub-arrays.) First developed by Tony Hoare in 1959, quicksort is still a commonly used algorithm for sorting in computer applications. On the average, it requires O(n log n) comparisons to sort n items, which compares favorably to other popular sorting methods, including merge sort and heapsort. Unfortunately, on rare occasions (including cases where the input is already sorted or contains items that are all equal) quicksort requires a worst-case O(n2) comparisons, while the other two methods remain O(n log n) in their worst cases. Still, when implemented well, quicksort can be about two or three times faster than its main competitors. Unlike merge sort, the standard implementation of quicksort does not preserve the order of equal input items (it is not stable), although stable versions of the algorithm do exist at the expense of requiring O(n) additional storage space. Other variations are based on different ways of choosing the pivot element (for example, choosing a random element instead of always using the last one), using more than one pivot, switching to an insertion sort when the sub-arrays have shrunk to a sufficiently small length, and using a three-way partitioning scheme (grouping items into those smaller, larger, and equal to the pivot—a modification that can turn the worst-case scenario of all-equal input values into the best case). Because of the algorithm's "divide and conquer" approach, parts of it can be done in parallel (in particular, the processing of the left and right sub-arrays can be done simultaneously). However, other sorting algorithms (including merge sort) experience much greater speed increases when performed in parallel.

Good articles - load new batch

Symbol support vote.svg
  These are Good articles, which meet a core set of high editorial standards.

  • Euclidean minimum spanning tree of 25 random points in the plane
    Euclidean minimum spanning tree of 25 random points in the plane
  • Vector addition and scalar multiplication: a vector v (blue) is added to another vector  w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.
    Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.
  • L6a4
    L6a4
  • Schematic representation of the Dirac delta by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.
    Schematic representation of the Dirac delta by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.
  • Image 5The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p − 1 is a prime number. The theorem is named after mathematicians Euclid and Leonhard Euler, who respectively proved the "if" and "only if" aspects of the theorem.It has been conjectured that there are infinitely many Mersenne primes. Although the truth of this conjecture remains unknown, it is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. However, it is also unknown whether there exists even a single odd perfect number. (Full article...)
    The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p − 1 is a prime number. The theorem is named after mathematicians Euclid and Leonhard Euler, who respectively proved the "if" and "only if" aspects of the theorem.

    It has been conjectured that there are infinitely many Mersenne primes. Although the truth of this conjecture remains unknown, it is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. However, it is also unknown whether there exists even a single odd perfect number. (Full article...)
  • In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.
    In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.
  • In this graph, an even number of vertices (the four vertices numbered 2, 4, 5, and 6) have odd degrees. The sum of degrees of all six vertices is (({1))}, twice the number of edges.
    In this graph, an even number of vertices (the four vertices numbered 2, 4, 5, and 6) have odd degrees. The sum of degrees of all six vertices is (({1))}, twice the number of edges.
  • A Penrose tiling with rhombi exhibiting fivefold symmetry
    A Penrose tiling with rhombi exhibiting fivefold symmetry
  • Image 9In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
  
    
      
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            1
            n
          
        
        =
        1
        +
        
          
            1
            2
          
        
        +
        
          
            1
            3
          
        
        +
        
          
            1
            4
          
        
        +
        
          
            1
            5
          
        
        +
        ⋯
        .
      
    
    {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n))=1+{\frac {1}{2))+{\frac {1}{3))+{\frac {1}{4))+{\frac {1}{5))+\cdots .}
  
The first 
  
    
      
        n
      
    
    {\displaystyle n}
  
 terms of the series sum to approximately 
  
    
      
        ln
        ⁡
        n
        +
        γ
      
    
    {\displaystyle \ln n+\gamma }
  
, where 
  
    
      
        ln
      
    
    {\displaystyle \ln }
  
 is the natural logarithm and 
  
    
      
        γ
        ≈
        0.577
      
    
    {\displaystyle \gamma \approx 0.577}
  
 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. (Full article...)
    In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:


    The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. (Full article...)
  • Francis Amasa Walker
    Francis Amasa Walker
  • Georg Cantor,     c. 1870
    Georg Cantor,     c. 1870
  • A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass
    A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass

Did you know (auto-generated) - load new batch

Nuvola apps filetypes.svg

More did you know – view different entries

Did you know...
Showing 7 items out of 75

Selected article – show another

A number is an abstract object that represents a count or measurement. A symbol for a number is called a numeral. The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

Numbers can be classified into sets called number systems. The most familiar numbers are the natural numbers, which to some mean the non-negative integers and to others mean the positive integers. In everyday parlance the non-negative integers are commonly referred to as whole numbers, the positive integers as counting numbers, symbolised by . Mathematics is used in many classes throughout the course of one's education.

The integers consist of the natural numbers (positive whole numbers and zero) combined with the negative whole numbers, which are symbolised by (from the German Zahl, meaning "number").

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face (for quotient). (Full article...)

View all selected articles

Subcategories

C Puzzle.png

Algebra | Arithmetic | Analysis | Complex analysis | Applied mathematics | Calculus | Category theory | Chaos theory | Combinatorics | Dynamical systems | Fractals | Game theory | Geometry | Algebraic geometry | Graph theory | Group theory | Linear algebra | Mathematical logic | Model theory | Multi-dimensional geometry | Number theory | Numerical analysis | Optimization | Order theory | Probability and statistics | Set theory | Statistics | Topology | Algebraic topology | Trigonometry | Linear programming


Mathematics | History of mathematics | Mathematicians | Awards | Education | Literature | Notation | Organizations | Theorems | Proofs | Unsolved problems

Full category tree. Select [►] to view subcategories.

Topics in mathematics

General Foundations Number theory Discrete mathematics
Nuvola apps bookcase.svg
Set theory icon.svg
Nuvola apps kwin4.png
Nuvola apps atlantik.png


Algebra Analysis Geometry and topology Applied mathematics
Arithmetic symbols.svg
Source
Nuvola apps kpovmodeler.svg
Gcalctool.svg

Index of mathematics articles

ARTICLE INDEX:
MATHEMATICIANS:

Related portals

WikiProjects

WikiProjects
The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion on the project's talk page.

In other Wikimedia projects

The following Wikimedia Foundation sister projects provide more on this subject:

More portals

Discover Wikipedia using portals