Math 2.0

Mathematical logic and foundations Theorem

Distributive law

Builds on Addition of natural numbersMultiplication of natural numbersMathematical induction

Statement

For all natural numbers aa, bb and cc, a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c. Here ++ is the addition and ⋅\cdot is the multiplication of natural numbers.

Remarks

This is the left distributive law. With commutativity of multiplication, it gives the right law (b+c)⋅a=b⋅a+c⋅a(b + c) \cdot a = b \cdot a + c \cdot a. The proof from the recursive definitions uses induction on cc. The same law holds in the integers, the rationals, the reals and the complex numbers.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Definition by recursion, concept, no evidence levelDefinition by recursi…conceptMathematical induction, concept, no evidence levelMathematical inductionconceptNatural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptSuccessor function, definition, no evidence levelSuccessor functiondefinitionAddition of natural numbers, definition, no evidence levelAddition of natural n…definitionMultiplication of natural numbers, definition, no evidence levelMultiplication of nat…definitionDistributive law, statement, no evidence levelDistributive lawstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Definition by recursion concept
  2. Mathematical induction concept
  3. Natural numbers object
  4. Peano axioms concept
  5. Successor function definition
  6. Addition of natural numbers definition
  7. Multiplication of natural numbers definition
  8. Distributive law statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Distributive_property, Definition
  2. Web https://en.wikipedia.org/wiki/Natural_number, Relationship between addition and multiplication
About this page
Tag0KBEMAJAYA
URIm2://0KBEMAJAYA
Aliasesdistributivity-of-multiplication-over-addition
Kindstatement, theorem
WritingW0 stub
MSC03F30
StatusThis copy of the site has no build data. Run m2 status 0KBEMAJAYA.
UsesAddition of natural numbers, Multiplication of natural numbers, Mathematical induction

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