Identity Matemathics
In mathematics and more generally in scientific fields, an identity mathematics is the finding that two mathematical objects (having two different mathematical scripts) are in fact the same object. In particular, an identity is an equality between two expressions which is true whatever the values of the different variables used. Identities are generally used to transform one mathematical expression into another, especially to solve an equation.
Identity Mathematics has several important uses:
- An identity is an equality that remains true even if you change all the variables that are used in that equality.^{}^{}
An equality in mathematical sense is only true under more particular conditions. For this, the symbol ≡ is sometimes used (note, however, that the same symbol can also be used for a congruence relation as well.)
- In algebra, an identity or identity element of a set S with an operation is an element which, when combined with any element s of S, produces s itself. In a group (an algebraic structure), this is often denoted by the symbol .^{}
- The identity function (or identity map) from a set S to itself, often denoted or , such that for all x in S.^{}
- In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is often denoted by the symbol
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Example Identity Mathematics
Identity relation
A common example of the first meaning is the trigonometric identity
which is true for all real values of θ (since the real numbers are the domain of both sine and cosine), as opposed to
- cosθ=1
which is only true for certain values of θ in a subset of the domain.
Identity element
The concepts of “additive identity” and “multiplicative identity” are central to the Peano axioms. The number 0 is the “additive identity” for integers, real numbers, and complex numbers. For the real numbers, for all
- 0 + a = a
- a + 0 = 0 and
- 0 + 0 = 0
Similarly, The number 1 is the “multiplicative identity” for integers, real numbers, and complex numbers. For the real numbers, for all
- 1 x a = a
- and
- 1 x 1 = 1
Identity function
A common example of an identity function is the identity permutation, which sends each element of the set {1, 2, …, n} itself.
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Common Identity Mathematics
Algebraic identities
Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra,^{} while other identities, such as (a+b)²=a²+2ab+b² and a²-b²=(a+b) (a-b), can be useful in simplifying algebraic expressions and expanding them.^{}
Trigonometric identities
Geometrically, trigonometric identities are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.
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 sin²θ + cos²θ = 1, which is true for all complex values of θ (since the complex numbers form the domain of sine and cosine). On the other hand, the equation
- cosθ=1
is only true for certain values of θ, not all (nor for all values in a neighborhood). For example, this equation is true when θ = 0, but false when θ = 2.
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity sin(2θ) = 2sinθcosθ, the addition formula for tan (x+y)),which can be used to break down expressions of larger angles into those with smaller constituents.
Exponential identities
The following identities hold for all integer exponents, provided that the base is non-zero:
Unlike addition and multiplication, exponentiation is not commutative. For example, 2 + 3 = 3 + 2 = 5 and 2 · 3 = 3 · 2 = 6, but 2^{3} = 8, whereas 3^{2} = 9.
And unlike addition and multiplication, exponentiation is not associative either. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9 and (2 · 3) · 4 = 2 · (3 · 4) = 24, but 2^{3} to the 4 is 8^{4} (or 4,096), whereas 2 to the 3^{4} is 2^{81} (or 2,417,851,639,229,258,349,412,352). Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up:
Logarithmic identities
Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another.^{}
Product, quotient, power and root
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the p-th power of a number is p times the logarithm of the number itself; the logarithm of a p-th root is the logarithm of the number divided by p. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions x = b^{logb(x)}, and/or y = b^{logb(y)}, in the left hand sides.
Formula | Example | |
---|---|---|
product | ||
quotient | ||
power | ||
root |
Change of base
The logarithm log_{b}(x) can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula:
Typical scientific calculators calculate the logarithms to bases 10 and e.^{} Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula:
Given a number x and its logarithm log_{b}(x) to an unknown base b, the base is given by:
Hyperbolic function identities
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn’s rule states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of 2, 6, 10, 14, … sinhs.^{}
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.
Logic and universal algebra
In mathematical logic and in universal algebra, an identity is defined as a formula of the form “∀x_{1},…,x_{n}. s = t“, where s and t are terms with no other free variables than x_{1},…,x_{n}. The quantifier prefix (“∀x_{1},…,x_{n}.”) is often left implicit, in particular in universal algebra. For example, the axioms of a monoid are often given as the identity set
- { ∀x,y,z. x*(y*z)=(x*y)*z , ∀x. x*1=x , ∀x. 1*x=x },
or, in short notation, as
- { x*(y*z)=(x*y)*z , x*1=x , 1*x=x }.
Some authors use the name “equation” rather than “identity”.
Remarkable identities: development and factorization
They are very useful for expanding or factoring literal expressions quickly.
You have to know them in both directions.
1) Square of a sum
(a + b) ² = a² + 2 × a × b + b²; also noted: (a + b) ² = a² + 2ab + b²
a² + b²: sum of squares
2 × a × b or 2ab: double product
Examples
Expansion: (3y + 1) ² = (3y) ² + 2 × 3y × 1 + 1² = 9y² + 6y + 1
Here the calculation is detailed but the goal is to succeed without the intermediate step.
Factoring: y² + 10y + 25 = y² + 2 × y × 5 + 5² = (y + 5) ²
Here, it’s the same, we must get there without the intermediate step.
2) Square of a difference
(a – b) ² = a² – 2ab + b²
Reminder: you don’t have to put a multiplied sign in front of a letter or parentheses
thus 2ab = 2 × a × b
Examples:
Development: (3y – 1) ² = 9y² – 6y + 1
Factoring: y² – 10y + 25 = (y – 5) ²
3) Product of the sum by the difference
(a + b) (a – b) = a² – b²
Pay attention to the ‘minus’ signs placed in the formula, I could very well have written the formula like this:
(-b + a) (b + a) = -b² + a² …
Examples:
Development: (3y + 1) (3y – 1) = 9y² – 1; (-5y + 1) (5y + 1) = – 25y² + 1 = 1 – 25y².
(-2y + 9) (2y – 9) this expression cannot be developed by the remarkable identity (a-b) (a + b) = a²-b² because there is a minus sign in each factor.
We can use the identity learned in 4th year not to forget:
(a + b) (c + d) = ac + ad + bc + bd.
Thus: (-2y + 9) (2y – 9) = -4y² + 18y + 18y – 81 = -4y² + 36y – 81.
We can also notice that the factor (-2y + 9) is the opposite of (2y – 9)
then we use the identity (a – b) ² = a² – 2ab + b².
So: (-2y + 9) (2y – 9) = – (2y-9) (2y-9) = – (4y²-36y + 81) = -4y² + 36y – 81.
Factorization: y² – 25 = y² – 5² = (y + 5) (y – 5);
4t² – (t-1) ² = (2t) ² – (t-1) ² = (2t- (t-1)) (2t + t-1) = (t + 1) (3t – 1)
Recognize the difference of two squares to factor using a² – b² = (a – b) (a + b)
There is no remarkable identity allowing to factorize a² + b² but be careful, if the given expression is of the type ‘-b² + a²’,
we could factor it because: -b² + a² = a² – b²
Examples, Questions and Answers for Identity Mathematics
Transform the following expressions using the remarkable identities: x and y are two real numbers: (5 + x) ²; (x-y) ²; 1-x²; y²-x²; (2 + y) ²; (x + y) ²; 16-x²; 49-x²; (8 + x) ²; 4-x²
Answer:
(5 + x) ² = 25 + 10x + x² = x² + 10x + 25
(x-y) ² = x²- 2x x y + y²
1-x² = Impossible, it is then factorization (1²-x²)
y²-x² = Impossible, it is factorization !!!
(2 + y) ² = 4 + 4y + y² = y² + 4y + 4
(x + y) ² = x² + 2x x y + y²
16-x² = Impossible, factorization (4²-x²)
49-x² = Same (7²-x²)
(8 + x) ² = 64 + 16x + x² = x² + 16x + 64
4-x² = Same, factorization (2²-x²)
Source: Wikipedia