In mathematics, a law is a formula that is always true within a given context.
[1] Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality,[2] or between formulas themselves, for instance, in mathematical logic.
is true for all real numbers a, and is therefore a law.
[4] Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena.
[5] The more significant laws are often called theorems.
, and the triangle inequality expresses a relationship between absolute values.
Pythagorean theorem: It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:[6] Geometrically, trigonometric identities are identities involving certain functions of one or more angles.
These identities are useful whenever expressions involving trigonometric functions need to be simplified.
Another important application is the integration of non-trigonometric functions: a common technique which involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
One of the most prominent examples of trigonometric identities involves the equation
), which can be used to break down expressions of larger angles into those with smaller constituents.
It is considered one of the most important and widely used inequalities in mathematics.
is always a non-negative real number (even if the inner product is complex-valued).
By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:[9][10] Moreover, the two sides are equal if and only if
[14] For example, of three gloves (none of which is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into.
De Morgan's laws: In propositional logic and Boolean algebra, De Morgan's laws,[15][16][17] also known as De Morgan's theorem,[18] are a pair of transformation rules that are both valid rules of inference.
They are named after Augustus De Morgan, a 19th-century British mathematician.
The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.
The rules can be expressed in English as: The three Laws of thought are: Benford's law is an observation that in many real-life sets of numerical data, the leading digit is likely to be small.
[21] In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time.
Uniformly distributed digits would each occur about 11.1% of the time.
[22] Strong law of small numbers, in a humorous way, states any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few.