completeness of the real numbers
(noun)
completeness implies that there are not any "gaps" or "missing points" in the real number line
Examples of completeness of the real numbers in the following topics:
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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 $\mathbb{Q}$.
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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.
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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).
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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.
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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
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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.
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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.
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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.
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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.
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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.