natural numbers

Examples of natural numbers in the following topics:

• Sequences of Mathematical Statements

  • The length of a sequence is the number of ordered elements, and it may be infinite.
  • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers.
  • For example, in the context of mathematical induction, a sequence of statements usually involves an algebraic statement into which you can substitute any natural number $(0, 1, 2, 3, ...)$ and the statement should hold true.

• Proof by Mathematical Induction

  • Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers (non-negative integers).
  • The simplest and most common form of mathematical induction proves that a statement involving a natural number $n$ holds for all values of $n$.
  • The choice between $n=0$ and $n=1$ in the base case is specific to the context of the proof: If $0$ is considered a natural number, as is common in the fields of combinatorics and mathematical logic, then $n=0$.
  • If, on the other hand, $1$ is taken as the first natural number, then the base case is given by $n=1$.

• The Number e

  • The number $e$ is an important mathematical constant, approximately equal to $2.71828$.
  • When used as the base for a logarithm, we call that logarithm the natural logarithm and write it as $\ln x$.
  • The number $e$, sometimes called the natural number, or Euler's number, is an important mathematical constant approximately equal to 2.71828.
  • When used as the base for a logarithm, the corresponding logarithm is called the natural logarithm, and is written as $\ln (x)$.
  • There are a number of different definitions of the number $e$.

• Sets of Numbers

  • The set of natural numbers, also known as "counting numbers," includes all whole numbers starting at 1 and then increasing.
  • The set of natural numbers is represented by the symbol $\mathbb{N}$ and can be denoted as $\mathbb{N}=\left \{ 1,2,3,4, \cdots \right \}$.
  • The set of real numbers includes every number, negative and decimal included, that exists on the number line.
  • The set of imaginary numbers, denoted by the symbol $\mathbb{I}$, includes all numbers that result in a negative number when squared.
  • The set of complex numbers, denoted by the symbol $\mathbb{C}$, includes a combination of real and imaginary numbers in the form of $a+bi$ where $a$ and $b$ are real numbers and $i$ is an imaginary number.

• Addition, Subtraction, and Multiplication

  • Complex numbers are added by adding the real and imaginary parts of the summands.
  • The multiplication of two complex numbers is defined by the following formula:
  • The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit.
  • Addition of two complex numbers can be done geometrically by constructing a parallelogram.
  • Discover the similarities between arithmetic operations on complex numbers and binomials

• Common Bases of Logarithms

  • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
  • The irrational number  $e\approx 2.718$ and arises naturally in financial mathematics in computations having to do with compound interest.
  • Natural logarithms are also used in physical sciences and pure math.
  • The entropy $(S)$ of a system can be calculated from the natural logarithm of the number of possible microstates $(W)$ the system can adopt:
  • Natural logarithms are closely linked to counting prime numbers ($2, 3, 5, 7$ ...), an important topic in number theory.

• Natural Logarithms

  • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
  • The natural logarithm is the logarithm with base equal to e.
  • The natural logarithm can be written as $\log_e x$ but is usually written as $\ln x$.
  • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts.
  • The first step is to take the natural logarithm of both sides:

• Logarithmic Functions

  • The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
  • Taking the logarithm of a number, one finds the exponent to which a certain value, known as a base, is raised to produce that number once more.
  • This is because the base $b$ is positive and raising a positive number to any power will yield a non-negative number.
  • A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$.
  • The irrational number  $e\approx 2.718$ arises naturally in financial mathematics, in computations having to do with compound interest and annuities.

• Converting between Exponential and Logarithmic Equations

  • While you can take the log to any base, it is common to use the common log with a base of $10$ or the natural log with the base of $e$.
  • This is because scientific and graphing calculators are equipped with a button for the common log that reads $log$, and a button for the natural log that reads $ln$ which allows us to obtain a good approximation for the common or natural log of a number.
  • Solve for $x$ in the equation $2^x=17$ using the natural log
  • Here we will use the natural logarithm instead to illustrate the fact that any base will do.

• Graphs of Logarithmic Functions

  • The domain of the function is all positive numbers.
  • The range of the function is all real numbers.
  • That is, the graph can take on any real number.
  • The range of the square root function is all non-negative real numbers, whereas the range of the logarithmic function is all real numbers.
  • When graphing with a calculator, we use the fact that the calculator can compute only common logarithms (base is $10$), natural logarithms (base is $e$) or binary logarithms (base is $2$).