- Any real number can be added to both sides.
- Any real number can be subtracted from both sides.
- Any real number can be multiplied to both sides.
- Any non-zero real number can divide both sides.
- Some functions can be applied to both sides. Caution must be exercised to ensure that the operation does not cause missing or extraneous solutions. For example, the equation has 2 sets of solutions: (with any x) and (with any y). Raising both sides to the exponent of 2 (which means, applying the function to both sides of the equation) changes our equation into, which not only has all the previous solutions but also introduces a new set of extraneous solutions, with and x being any number.
The algebraic properties (1-4) imply that equality is a congruence relation for a field; in fact, it is essentially the only one.
The most well known system of numbers which allows all of these operations is the real numbers, which is an example of a field. However, if the equation were based on the natural numbers for example, some of these operations (like division and subtraction) may not be valid as negative numbers and non-whole numbers are not allowed. The integers are an example of an integral domain which does not allow all divisions as, again, whole numbers are needed. However, subtraction is allowed, and is the inverse operator in that system.
If a function that is not injective is applied to both sides of a true equation, then the resulting equation will still be true, but it may be less useful. Formally, one has an implication, not an equivalence, so the solution set may get larger. The functions implied in properties (1), (2), and (4) are always injective, as is (3) if we do not multiply by zero. Some generalized products, such as a dot product, are never injective.For more details on this topic, see Equation solving.
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Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)