completeness of the real numbers

Examples of completeness of the real numbers in the following topics:

• Intermediate Value Theorem

  • For a real-valued continuous function $f$ on the interval $[a,b]$ and a number $u$ between $f(a)$ and $f(b)$, there is a $c \in [a,b]$ such that $f(c)=u$.
  • Version I: If $f$ is a real-valued continuous function on the interval $[a, b]$, and $u$ is a number between $f(a)$ and $f(b)$, then there is a $c \in [a, b]$ such that $f(c) = u$.
  • Version 3: Suppose that $I$ is an interval $[a, b]$ in the real numbers $\mathbb{R}$ and that $f : I \to R$ is a continuous function.
  • The theorem depends on (and is actually equivalent to) the completeness of the real numbers.
  • It is false for the rational numbers $\mathbb{Q}$.

• The Intermediate Value Theorem

  • Stated in the language of algebra supported by : If f is a real-valued continuous function on the interval [a, b], and u is a number between f(a) and f(b), then there is a c ∈ [a, b] such that f(c) = u.
  • It is frequently stated in the following equivalent form: Suppose that f : [a, b] → R is continuous and that u is a real number satisfying f(a) < u < f(b) or f(a) > u > f(b).
  • The theorem depends on (and is actually equivalent to) the completeness of the real numbers.
  • It is false for the rational numbers Q.
  • The function is defined for all real numbers x ≠ −2 and is continuous at every such point.

• The Fundamental Theorem of Algebra

  • However, despite its name, no purely algebraic proof exists, since every proof makes use of the fact that $\mathbb{C}$ is complete.
  • In particular, since every real number is also a complex number, every polynomial with real coefficients does admit a complex root.
  • The complex conjugate root theorem says that if a complex number $a+bi$ is a zero of a polynomial with real coefficients, then its complex conjugate $a-bi$ is also a zero of this polynomial.
  • This last remark, together with the alternative statement of the fundamental theorem of algebra, tells us that the parity of the real roots (counted with multiplicity) of a polynomial with real coefficients must be the same as the parity of the degree of said polynomial.
  • Therefore, a polynomial of even degree admits an even number of real roots, and a polynomial of odd degree admits an odd number of real roots (counted with multiplicity).

• Introduction to Complex Numbers

  • In this expression, $a$ is called the real part and $b$ the imaginary part of the complex number.
  • In this way, the set of ordinary real numbers can be thought of as a subset of the set of complex numbers.
  • It is beneficial to think of the set of complex numbers as an extension of the set of real numbers.
  • This extension makes it possible to solve certain problems that can't be solved within the realm of the set of real numbers.
  • has no solution if we restrict ourselves to the real numbers, since the square of a real number is never negative.

• Addition and Subtraction of Complex Numbers

  • For example, the sum of $2+3i$ and $5+6i$ can be calculated by adding the two real parts $(2+5)$ and the two imaginary parts $(3+6)$ to produce the complex number $7+9i$.
  • Note that this is always possible since the real and imaginary parts are real numbers, and real number addition is defined and understood.
  • As another example, consider the sum of $1-3i$ and $4+2i$.
  • However, two real numbers can never add to be a non-real complex number.
  • Calculate the sums and differences of complex numbers by adding the real parts and the imaginary parts separately

• Interval Notation

  • A "real interval" is a set of real numbers such that any number that lies between two numbers in the set is also included in the set.
  • Other examples of intervals include the set of all real numbers and the set of all negative real numbers.
  • The two numbers are called the endpoints of the interval.
  • The set of all real numbers is the only interval that is unbounded at both ends; the empty set (the set containing no elements) is bounded.
  • Representations of open and closed intervals on the real number line.

• Real Numbers, Functions, and Graphs

  • Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced.
  • The real numbers include all the rational numbers, such as the integer -5 and the fraction $\displaystyle \frac{4}{3}$, and all the irrational numbers such as $\sqrt{2}$ (1.41421356… the square root of two, an irrational algebraic number) and $\pi$ (3.14159265…, a transcendental number).
  • Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one.
  • The real line can be thought of as a part of the complex plane, and correspondingly, complex numbers include real numbers as a special case.
  • Here, the domain is the entire set of real numbers and the function maps each real number to its square.

• Absolute Value

  • Absolute value can be thought of as the distance of a real number from zero.
  • In mathematics, the absolute value (sometimes called the modulus) of a real number $a$ is denoted $\left | a \right |$.
  • For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5, because both numbers are the same distance from 0.
  • When applied to the difference between real numbers, the absolute value represents the distance between the numbers on a number line.
  • The absolute values of 5 and -5 shown on a number line.

• Round-off Error

  • A round-off error is the difference between the calculated approximation of a number and its exact mathematical value.
  • A round-off error, also called a rounding error, is the difference between the calculated approximation of a number and its exact mathematical value.
  • Numerical analysis specifically tries to estimate this error when using approximation equations, algorithms, or both, especially when using finitely many digits to represent real numbers.
  • The more digits that are used, the more accurate the calculations will be upon completion.
  • The number $\pi$ (pi) has infinitely many digits, but can be truncated to a rounded representation of as 3.14159265359.

• Factors

  • This process has many real-life applications and can help us solve problems in mathematics.
  • To find the factors, consider the numbers that yield a product of 24.
  • This is a complete list of the factors of 24.
  • In a factor tree, the number of interest is written at the top.
  • This process repeats for each subsequent factor of the original number until all the factors at the bottoms of the branches are prime.