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An overview of real numbers, including their representation, classification, and ordering. It covers topics such as the subsets of real numbers (natural numbers, whole numbers, integers), the distinction between rational and irrational numbers, the use of the real number line, and the properties of ordering real numbers using inequalities. The document also introduces the concepts of absolute value and distance between real numbers, as well as the basic rules and properties of algebra, including evaluating algebraic expressions, the four arithmetic operations, and the fundamental theorem of arithmetic. This comprehensive coverage of real numbers and algebraic concepts would be useful for students studying mathematics, particularly in the context of foundational courses in algebra, precalculus, or calculus.
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Real numbers are used in everyday life to describe quantities such as age, miles per gallon, and population. Real numbers are represented by symbols such as
Here are some important subsets (each member of subset B is also a member of set A) of the set of real numbers. {1, 2, 3, 4,.. .} {0, 1, 2, 3, 4,.. .} {... , –3, –2, –1, 0, 1, 2, 3,.. .} Set of natural numbers Set of whole numbers Set of integers
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Real numbers are represented graphically by a real number line. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive and numbers to the left of 0 are negative, as shown in Figure P.2. The term nonnegative describes a number that is either positive or zero. Figure P. The Real Number Line Origin Negative direction Positive direction 8
There is a one-to-one correspondence between real numbers and points on the real number line. That is, every point on the real number line corresponds to exactly one real number, called its coordinate, and every real number corresponds to exactly one point on the real number line, as shown in Figure P.3. Figure P. One-to-One Correspondence Every real number corresponds to exactly one point on the real number line.
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One important property of real numbers is that they are ordered. Geometrically, this definition implies that a < b if and only if a lies to the left of b on the real number line, as shown in Figure P.4. Figure P. a < b if and only if lies to the left of b.
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Inequalities can be used to describe subsets of real numbers called intervals. In the bounded intervals below, the real numbers a and b are the endpoints of each interval. 14
The symbols , positive infinity, and , negative infinity, do not represent real numbers. They are simply convenient symbols used to describe the unboundedness of an interval, such as (1, ) or ( , 3].
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Use inequality notation to represent each of the following. a. c is at most 2. b. All x in the interval (–3, 5]. c. t is at least 4 but less than 11. Solution: a. The statement “c is at most 2” can be represented by c 2. 16
b. “All x in the interval (–3, 5]” can be represented by
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The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a = b, a < b, or a > b. (^) Law of Trichotomy 20
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The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. 22
Notice from this definition that the absolute value of a real number is never negative. For instance, if a = –5, then |–5| = –(–5) = 5. The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0. So, |0| = 0.
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One characteristic of algebra is the use of letters to represent numbers. The letters are variables, and combinations of letters and numbers are algebraic expressions. Here are a few examples of algebraic expressions. 5 x, 2 x – 3, , 7 x + y 28
The terms of an algebraic expression are those parts that are separated by addition. For example, x^2 – 5x + 8 = x^2 + (–5x) + 8 has three terms: x^2 and –5x are the variable terms, and 8 is the constant term. The numerical factor of a term is called the coefficient. For instance, the coefficient of –5x is –5, and the coefficient of x^2 is 1. To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression.
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Subtraction and division are the inverse operations of addition and multiplication, respectively. Subtraction: Add the opposite of b. a – b = a + (–b) Division: Multiply by the reciprocal of b. If b ≠ 0, then a/b. 32
In these definitions, –b is the additive inverse (or opposite) of b, and 1/b is the multiplicative inverse (or reciprocal) of b. In the fractional form a/b, a is the numerator of the fraction and b is the denominator.
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Because the properties of real numbers below are true for variables and algebraic expressions, as well as for real numbers, they are often called the Basic Rules of Algebra. Try to formulate a verbal description of each property. For instance, the Commutative Property of Addition states that the order in which two real numbers are added does not affect their sum. 34
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a. b. Divide fractions: Equivalent fractions:
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If a, b, and c are integers such that ab = c, then a and b are factors or divisors of c. A prime number is an integer that has exactly two positive factors: itself and 1. For example, 2, 3, 5, 7, and 11 are prime numbers. The numbers 4, 6, 8, 9, and 10 are composite because they can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite. 40
The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as the product of prime numbers. For instance, the prime factorization of 24 is 24 = 2 2 2 3.
