Distributivity

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In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:

In the left-hand side of the above equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the results added afterwards. Because these give the same final answer (8), we say that multiplication by 2 distributes over addition of 1 and 3. Since we could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Contents

Definition

Given a set S and two binary operations · and + on S, we say that the operation ·

  • is left-distributive over + if, given any elements x, y, and z of S,
  • is right-distributive over + if, given any elements x, y, and z of S:
  • is distributive over + if it is both left- and right-distributive.[1]

Notice that when · is commutative, then the three above conditions are logically equivalent.

Examples

Distributivity and rounding

In practice, the distributive property of multiplication (and division) over addition is lost around the limits of arithmetic precision. For example, the identity ⅓+⅓+⅓ = (1+1+1)/3 appears to fail if conducted in decimal arithmetic; however many significant digits are used, the calculation will take the form 0.33333+0.33333+0.33333 = 0.99999 ≠ 1. Even where fractional numbers are representable exactly, errors will be introduced if rounding too far; for example, buying two books each priced at £14.99 before a tax of 17.5% in two separate transactions will actually save £0.01 over buying them together: £14.99×1.175 = £17.61 to the nearest £0.01, giving a total expenditure of £35.22, but £29.98×1.175 = £35.23. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.

Distributivity in rings

Distributivity is most commonly found in rings and distributive lattices.

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