Real numbers

We use numbers to measure and compare quantities. The types of numbers that make up the real number system include:

Natural numbers

Integers, consisting natural numbers with their negatives and 0

Rational numbers, ratios of integers ()

Irrational numbers, which cannot be expressed as a ratio of integers

The set of all real numbers is denoted by the symbol . A rational number’s corresponding decimal is repeating. A irrational number’s decimal representation is nonrepeating. For example,

The bar indicates the sequence of digits repeats indefinitely.

Stop the decimal expression of a number at a certain place, and we get an approximation:

To convert a rational number’s repeating decimal such as to a ratio of two integers, multiply it by appropriate powers of 10 and subtract to eliminate the repeating part:

Thus .

Properties of Real Numbers

Commutative Properties

Order doesn’t matter when adding two numbers:

Order doesn’t matter when multiplying two numbers:

Associative Properties

It doesn’t matter which two numbers to add first when adding three numbers:

It doesn’t matter which two numbers to multiply first when multiplying three numbers:

Distributive Property

Multiplying a number by a sum is the same as multiplying the number by each term and add the results:

Addition and Subtraction

is the additive identity(加法单位元) for addition, as for any real number . Subtraction is the operation that undoes addition; to subtract a number from another is to add the negative of that number:

Multiplication and Division

is the multiplicative identity(乘法单位元) as for any real number . Every nonzero real number has an inverse which satisfies . Division is the operation that undoes multiplication; to divide by a number is to multiply the inverse of that number:

It’s written simply as . We refer to it as the quotient(商) of and or as the fraction over ; is the numerator and is the denominator(or divisor).

Properties of Fractions

  1. Multiply both numerators and denominators when multiplying fractions:
  1. Invert the divisor and multiply it when dividing fractions:
  1. Add the numerators when adding fractions with the same denominator:
  1. Find a common denominator then add the numerators when adding fractions with different denominators:
  1. Cancel numbers that are common factors:

Note that we don’t usually use Property 4, and instead, we find the smallest possible common denominator–the Least Common Denominator(LCD) and then use Property 3.

Example: finding the LCD

Evaluate: Solution: find the LCD by factoring each denominator into prime factors by trial division, then form the product of the highest power of each prime factors. For this example, we have:

Thus the LCD is .

The Real Line

Real numbers can be represented by points on a line. We choose an arbitrary reference point called the origin, which corresponds to , and, any unit of measurement. Each number is represented by the point on the line a distance of units to the origin. The number associated with a point is called its coordinate, and the line is called a coordinate line or a real number line, or simply a real line. The real numbers are ordered.

Sets and Intervals

A set is a collection of objects which are called elements of the set. Notation means is an element of set and is not. If set A consists of all positive integers less than 7, then it can be written as

or, in set-builder notation as

The union of set and , written as , is a set that consists of all elements that are in and . Their intersection(交) consists of elements that are in both and .

The empty set, denoted by , contains no elements.

Intervals are sets of real numbers. The open interval from to consists of all numbers between and and is denoted by . The closed interval from to includes the endpoints and is denoted by .

Absolute Value

The absolute value of a number , denoted by , is the distance from to on the real line, which is always positive or zero. By definition,

The distance between the points and is .

Properties of Absolute Value

A number’s absolute value is always positive or zero:

A number and its negative have the same absolute value:

The absolute value of a product equals the product of absolute values:

Triangle Inequality:

Exponents and Radicals

A product of identical numbers is written in exponential notation, for example, as . By definition, for any real number and positive integer , the nth power of is

The number is called the base, is called the exponent.

Laws of Exponents

To multiply two powers of the same base, add the exponents:

To make and negative integers follow the same rule, zero and negative exponents are defined as

To divide two powers of the same base, subtract the exponents:

To raise a power to a new power, multiply the exponents:

To raise a product to a power, raise each factor to the power:

To raise a quotient to a power, raise both numerator and denominator to the power:

Change the sign of the exponent when moving a number raised to a power between numerator and denominator:

Scientific Notation

A positive number is said to be written in scientific notation if it’s expressed as follows:

The exponent indicates how many places the decimal point should be moved. For instance,

The positive exponent indicates the decimal point should be moved 6 places to the right, and for

The negative exponent indicates the decimal point should be moved 3 places to the left.

Radicals

The symbol means “the positive square root of”(also called the principle square root of), thus

Because , the symbol makes sense only when . Similarly, if is any positive integer, the principal th root of a is defined as follows:

If is even, then we must have and .

Rational Exponents

To give meaning to the symbol in a way that’s consistent with the laws of exponents, we would have to have

So by definition of th root,

In general, rational exponents are defined as follows: