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Equality (mathematics)
An equation is simply an assertion that two expressions are related by equality (are equal) (=).
The etymology of the word equality is from the Latin aequalis, meaning uniform or identical, from aequus, meaning "level, even, or just."
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Substitution Property of Equality[edit]
The substitution property makes it possible, among other things, to manipulate elements in an equation so that, for example, value(s) of individual elements, previously unknown (and therefore referred to as letters), can be determined.
- For any[1] quantities a and b and any expression F(x), If[2] a = b, then F(a) = F(b) (if either side makes sense, i.e. is well-formed).
Some specific examples of this are:
- For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);
- For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);
- For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);
- For any real numbers a, b, and c, if a = b And[3] c Is not[4] zero, then a/c = b/c (here F(x) is x/c).
One of the primary practical uses of these equation is to find the value of a particular element in an equation.
Practical example[edit]
Multiplying both sides of an equation by the divisor number on one side to eliminate it is the example
The rest of it is just to illustrate its practical application
See Earth's Orbit
Imperical observations:The Earth is at its closest to the Sun on Jan 2nd-5th, and at its furthest away on July 2nd-5th. There is some effect of the distance between the sun and the earth on the radiant energy reaching the earth.
Deduction (or Induction, depending on how strictly the definitions are applied): Therefore, there is some measure of additional radiant energy up to a maximum at Jan 2nd-5th, tending to heat northern latitudes up in their late midwinter, and heating southern latitudes up in their late midsummer. (Deduction again, last warning): Thus, the winters in northern latitudes (Removing Variables to make the hypothesis as true as possible): All other variables such as warming by ocean currents, etc, removed from consideration[5]), are relatively more mild, and the summers in southern latitudes (variables removed), relatively more extreme. However, the ratio of the Wikipedia:aphelion (furthest distance) to the Wikipedia:perihelion (nearest) is only 1.033988391 : 1, a difference of 3.4%.
Hypothesis: IF the difference is 3.4%, AND Inverse-square law says, "the radiation passing through any unit area is inversely proportional to the square of the distance from the point source" (received light quadruples as the distance halves) THEN We can calculate how much more radiant energy is added to Earth when it is closest to the sun than when it is furthest away:
Inverse-square Law (science): E = I / d 2
E, sun energy is a constant. It stays the same no matter where the Earth is, so can be made equal to 1. Trust me, you want as many 1s as possible.
We are solving for I, light intensity on Earth.
d is relative distance. Perihelion distance is a constant: 1 (yay). Comparing aphelion distance to this gives d.
The ratio of the aphelion to the perihelion is 1.033988391 : 1
Therefore, d = 1.033988391
1 = I / 1.0339883912
d2 = (1.033988391 times 1.033988391) = 1.069131994
1 = I / 1.069131994
Multiply both sides of the equation by 1.069131994
We can do this and keep the integrity of the equation, because of the substitution property
1.069131994 = I divided by 1.069131994, times 1.069131994
I is now both multiplied and divided by the same number, 1.069131994. (any number both divided by and multiplied by the same number = that number; these cancel each other out)
1.069131994 = I
E is now 1.069131994, and now equal to I, as it is no longer divided by anything. #Symmetric property says that, therefore, I = 1.069131994
Conclusion
1.069 times (aphelion energy) = perihelion energy. 7% more light intensity at perihelion than at aphelion. Perihelion intensity is 107% of aphelion. The Earth as a whole gets 7% more light energy on January 2nd-5th than it does on July 2nd-5th.
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Related Properties[edit]
Reflexive property[edit]
The reflexive property states:
- For any quantity a, a = a.
This property is generally used in mathematical proofs as an intermediate step.
Symmetric property[edit]
The symmetric property states:
- For any quantities a and b, If a = b, then b = a.
Transitive property[edit]
The transitive property states:
Mathematical definitions[edit]
Loosely, equality is the state of being quantitatively the same. The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation on S that is both an equivalence relation and a partial order. It follows from this that equality is the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence relation on S.
Logical formulations[edit]
The equality relation is always defined such that things that are equal have all and only the same properties. Some people define equality as congruence. Often equality is just defined as identity.
A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally:
- Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).
In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.
Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems. If a=b, then a can replace b and b can replace a.
Relation with equivalence and isomorphism[edit]
- See also: Equivalence relation and Isomorphism
In some contexts, equality is sharply distinguished from equivalence or isomorphism.[7] For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions 1/2 and 2/4 are distinct as fractions, as different strings of symbols, but they "represent" the same rational number, the same point on a number line. This distinction gives rise to the notion of a quotient set.
Similarly, the sets { A, B, C } and { 1, 2, 3 }
are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements, and thus isomorphic, meaning that there is a bijection between them, for example
- A |-> 1, B |-> 2, C |-> 3
However, there are other choices of isomorphism, such as
- A |-> 3, B |-> 2 C |-> 1
and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of fundamental importance in category theory, and is one motivation for the development of category theory.
See Also[edit]
References[edit]
- ↑ (Universal quantification)
- ↑ (Material implication)
- ↑ (And (logic))
- ↑ (Division by zero)
- ↑ England, for example, is warmed by Atlantic ocean currents, and this warms it more than proximity to the sun. It is at the same latitude as Canada, but it is generally warmer. Both, however, tend to be warmer in the winter than an equivalent southern latitude location
- ↑ (Wikipedia:And (logic))
- ↑ Template:Harv
