# Distributivity

 related topics {math, number, function} {law, state, case}

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.

### 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.