original number. Addition is commutative and associative, so the order in which finitely many terms are added does not matter. The number has the property that, when added to any number, it yields that same number; so, it is the identity element of addition, or the additive identity. For every number , there is a number denoted , called the opposite of , such that and . So, the opposite of is the inverse of with respect to addition, or the additive inverse of . For example, the opposite of is , since . Addition can also be interpreted geometrically, as in the following example. If we have two sticks of lengths 2 and 5, then, if the sticks are aligned one after the other, the length of the combined stick becomes 7, since . Subtraction Subtraction, denoted by the symbol , is the inverse operation to addition. Subtraction finds the difference between two numbers, the minuend minus the subtrahend: Resorting to the previously established addition, this is to say that the difference is the number that, when added to the subtrahend, results in the minuend: For positive arguments and holds: If the minuend is larger than the subtrahend, the difference is positive. If the minuend is smaller than the subtrahend, the difference is negative. In any case, if minuend and subtrahend are equal, the difference Subtraction is neither commutative nor associative. For that reason, the construction of this inverse operation in modern algebra is often discarded in favor of introducing the concept of inverse elements (as sketched under ), where subtraction is regarded as adding the additive inverse of the subtrahend to the minuend, that is, . The immediate price of discarding the binary operation of subtraction is the introduction of the (trivial) unary operation, delivering the additive inverse for any given number, and losing the immediate access to the notion of difference, which is potentially misleading when negative arguments are involved. For any representation of numbers, there are methods for calculating results, some of which are particularly advantageous in exploiting procedures, existing for one operation, by small alterations also for others. For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware (negation). The trade-off is the halving of the number range for