Rational Numbers As Exponents by tutorciecle123


									           Rational Numbers As Exponents
Rational Numbers As Exponents

We can represent Rational Numbers as exponents, but before representing we need to know
What is a Rational Number, any number that can be represented in the a/b, where a and b are
real numbers, where b can't be zero.

We can perform many operations in rational number as addition, subtraction, division and
multiplication. Rational numbers have a wide application in the world of mathematics. If we
talk about exponential functions, they are always represented by e^x.

These functions are generally used in Integration and differentiation; these types of functions
are always increasing. We can represent them in Logarithm form as well.

As we are seeing the general form of exponential function is e^x so when x will increase then
surely function will increase. Now we will see how we can represent rational number as
exponent, we can use rational number in place of x, so we can write the exponential function
as e^3/4.

Now we will see how we can multiply two exponential functions when you have rational
numbers in their powers.
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In mathematics, a rational number is any number that can be expressed as the quotient or
fraction a/b of two integers, with the denominator b not equal to zero.

Since b may be equal to 1, every integer is a rational number. The set of all rational numbers
is usually denoted by a boldface Q (or blackboard bold , Unicode ℚ), which stands for

The decimal expansion of a rational number always either terminates after a finite number of
digits or begins to repeat the same finite sequence of digits over and over.

Moreover, any repeating or terminating decimal represents a rational number. These
statements hold true not just for base 10, but also for binary, hexadecimal, or any other
integer base.

A real number that is not rational is called irrational. Irrational numbers include √2, π, and e.
The decimal expansion of an irrational number continues forever without repeating. Since the
set of rational numbers is countable, and the set of real numbers is uncountable, almost all
real numbers are irrational.

The rational numbers can be formally defined as the equivalence classes of the quotient set
(Z × (Z ∖ {0})) / ~, where the cartesian product Z × (Z ∖ {0}) is the set of all ordered pairs (m,n)
where m and n are integers, n is not zero (n ≠ 0), and "~" is the equivalence relation defined
by (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0.

In abstract algebra, the rational numbers together with certain operations of addition and
multiplication form a field. This is the archetypical field of characteristic zero, and is the field of
fractions for the ring of integers.

Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the
field of algebraic numbers.

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In mathematical analysis, the rational numbers form a dense subset of the real numbers. The
real numbers can be constructed from the rational numbers by completion, using Cauchy
sequences, Dedekind cuts, or infinite decimals.

The equivalence relation (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0 is a congruence
relation, i.e. it is compatible with the addition and multiplication defined above, and we may
define Q to be the quotient set (Z × (Z ∖ {0})) / ∼, i.e.

we identify two pairs (m1,n1) and (m2,n2) if they are equivalent in the above sense. (This
construction can be carried out in any integral domain: see field of fractions.)

We denote by [(m1,n1)] the equivalence class containing (m1,n1). If (m1,n1) ~ (m2,n2) then,
by definition, (m1,n1) belongs to [(m2,n2)] and (m2,n2) belongs to [(m1,n1)]; in this case we
can write [(m1,n1)] = [(m2,n2)]

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