Inverse

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Inverse has a number of closely related meanings both in everyday language and more specialized contexts, particularly in mathematics.

Generally an inverse is the reverse or opposite of a quantity, procedure, relationship or effect.

In mathematics the exact meaning of inverse may be determined by another word - for example an additive inverse is the number which when added to the original number makes zero, so -3 is the additive inverse of 3. At other times its meaning may be determined by context - when discussing functions an inverse will be understood to be the function that reverses the effect of a given function, so for non-negative real numbers taking the positive square root would be the inverse of squaring.

The concept of inverses is strongly related to that of "identity". A mathematical object, when combined with its inverse, will produce the relevant identity. In the examples above zero is the identity for addition, and the function which leaves things unchanged is called the identity function.

Examples

  • The additive inverse of a number is the number which when added to the original number makes 0, the additive identity. The inverse of positive numbers is simply their negative, of negative numbers similarly their corresponding positive.
  • The multiplicative inverse of a number is the number which when multiplied to the original number makes 1, the multiplicative identity. Another word for multiplicative inverse is reciprocal. The multiplicative inverse is equal to 1 divided by that number, so the multiplicative inverse of 4 is 1/4 or 0.25.
  • The inverse of a square matrix is the matrix which when multiplied by the original makes the identity matrix I, that is the matrix with 1s on its main diagonal and 0s elsewhere. Not all square matrices have an inverse and those that don't are called singular. Determining an inverse of a matrix is a laborious task of arithmetic, particularly for larger matrices, and ideally suited to computers.
  • The inverse of a function is a function that reverses the action of the original function. Not all functions are invertible; for example the square of both -3 and 3 is 9, so if we were to reverse this, by square rooting, we would have to decide whether to return from 9 back to the 3 or the -3. This problem can be solved by limiting the domain of the original function (in our example to non-negative numbers). Composing a function with its inverse yields the identity function.
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