Lecture

Intervals: Real Numbers

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Description

This lecture covers the concept of intervals in real numbers, including open, closed, and half-open intervals, as well as periodic decimal numbers and irrational numbers. The instructor explains the notation and properties of intervals, emphasizing the distinction between rational and irrational numbers.

In MOOCs (2)

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

Analysis I (part 1): Prelude, basic concepts, real numbers

Can you calculate the value of sin(π/4)? Does the identity ln(xy)=ln(x)+ln(y) sound familiar? In the prelude we recall and expand on a few such notions, in particular the trigonometric functions sin, cos and tan and their properties, the concept of reciprocal functions such as exp and ln, some calculation rules concerning powers, logarithms and roots, sets and functions. Next we discuss number systems: based on the "intuitive" notion of the natural integers N={0,1,2,3,...}, we rigorously define the rational

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Related concepts (32)

Nested intervals

In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: Every interval in the sequence is contained in the previous one ( is always a subset of ). The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ).

Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0, 1, and all numbers in between. Other examples of intervals are the set of numbers such that 0 < x < 1, the set of all real numbers , the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element).

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction \tfrac p q of two integers, a numerator p and a non-zero denominator q. For example, \tfrac{-3}{7} is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold \Q. A rational number is a real number.

Construction of the real numbers

In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition. The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them.

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