Operations with Real Numbers

If a, b, c belong to the set R of real numbers, then:

	a + b and ab belong to R	Closure law
	. a + b = b + a	Commutative law of addition
	a + (b + c) = (a + b) + c	Associative law of addition
	ab = ba	Commutative law of multiplication
	a(bc) = (ab)c	Associative law of multiplication
	a(b + c) = ab + ac	Distributive law
	a + 0 = 0 + a = a, 1 ・ a = a ・ 1 = a

Inequalities

If a – b is a nonnegative number, we say that a is greater than or equal to b or b is less than or equal to a, and write, respectively, a > b or b < a. If there is no possibility that a = b, we write a > b or b < a. Geometrically, a > b if the point on the real axis corresponding to a lies to the right of the point corresponding to b.

Absolute Value of Real Numbers

The absolute value of a real number a, denoted by |a|, is defined as a if a > 0, – a if a < 0, and 0 if a = 0.

The distance between any two points (real numbers) a and b on the real axis is |a–b| = |b–a|.

Exponents and Roots

The product a・ a . . . a of a real number a by itself p times is denoted by a^p, where p is called the exponent and a is called the base.

If a^p = N, where p is a positive integer, we call a a pth root of N, written . There may be more than one real pth root of N.

If p and q are positive integers, we define .

Logarithms

If a^p = N, p is called the logarithm of N to the base a, written p = log_a N. If a and N are positive and a # 1, there is only one real value for p. In practice, two bases are used: base a = 10, and the natural base a = e = 2.71828. . . . The logarithmic systems associated with these bases are called common and natural, respectively. The common logarithm system is signified by log N; i.e., the subscript 10 is not used. For natural logarithms, the usual notation is ln N.

Point Sets, Intervals

A set of points (real numbers) located on the real axis is called a one-dimensional point set.

The set of points x such that a £ x£ b is called a closed interval and is denoted by [a, b]. The set a < x < b is called an open interval, denoted by (a, b). The sets a < x £ b and a £ x < b, denoted by (a, b] and [a, b), respectively, are called half-open or half-closed intervals.

The symbol x, which can represent any number or point of a set, is called a variable. The given numbers a or b are called constants.

Letters were introduced to construct algebraic formulas around 1600. Not long thereafter, the philosopher-mathematician Rene Descartes suggested that the letters at the end of the alphabet be used to represent variables and those at the beginning to represent constants. This was such a good idea that it remains the custom.

EXAMPLE. The set of all x such that |x| < 4, i.e., –4 < x < 4, is represented by (–4, 4), an open interval.

The set x > a can also be represented by a < x < ¥_. Such a set is called an infinite or unbounded interval. Similarly, –_¥ < x < ¥ _ represents all real numbers x.

Countability

A set is called countable or denumerable if its elements can be placed in 1-1 correspondence with the natural numbers.

EXAMPLE. The even natural numbers 2, 4, 6, 8, . . . is a countable set because of the 1-1 correspondence shown.

A set is infinite if it can be placed in 1-1 correspondence with a subset of itself. An infinite set which is countable is called countable infinite.

The set of rational numbers is countable infinite, while the set of irrational numbers or all real numbers is noncountably infinite.

The number of elements in a set is called its cardinal number. A set which is countably infinite is assigned the cardinal number À₀ (the Hebrew letter aleph-null). The set of real numbers (or any sets which can be placed into 1-1 correspondence with this set) is given the cardinal number C, called the cardinality of the contimuum.

Neighborhoods

The set of all points x such that |x – a|< δ, where δ > 0, is called a δ neighborhood of the point a. The set of all points x such that 0 < |x – a| < δ, in which x = a is excluded, is called a deleted δ neighborhood of a or an open ball of radius δ about a.

Limit Points

A limit point, point of accumulation, or cluster point of a set of numbers is a number l such that every deleted δ neighborhood of l contains members of the set; that is, no matter how small the radius of a ball about l, there are points of the set within it. In other words, for any δ > 0, however small, we can always find a member x of the set which is not equal to l but which is such that |x – l| < δ. By considering smaller and smaller values of δ, we see that there must be infinitely many such values of x.

A finite set cannot have a limit point. An infinite set may or may not have a limit point. Thus, the natural numbers have no limit point, while the set of rational numbers has infinitely many limit points.

A set containing all its limit points is called a closed set. The set of rational numbers is not a closed set, since, for example, the limit point is not a member of the set. However, the set of all real numbers x such that 0 < x < 1 is a closed set.

Bounds

If for all numbers x of a set there is a number M such that x < M, the set is bounded above and M is called an upper bound. Similarly if x > m, the set is bounded below and m is called a lower bound. If for all x we have m < x < M, the set is called bounded.

If M is a number such that no member of the set is greater than M but there is at least one member which exceeds M– e for every e > 0, then M is called the least upper bound (l.u.b.) of the set. Similarly, if no member of the set is smaller than m +e for every e> 0, then m is called the greatest lower bound (g.l.b.) of the set.