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
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
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
Multiplication and Division
It’s written simply as
Properties of Fractions
- Multiply both numerators and denominators when multiplying fractions:
- Invert the divisor and multiply it when dividing fractions:
- Add the numerators when adding fractions with the same denominator:
- Find a common denominator then add the numerators when adding fractions with different denominators:
- 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:
Thus the LCD is
The Real Line
Real numbers can be represented by points on a line. We choose an arbitrary reference point
Sets and Intervals
A set is a collection of objects which are called elements of the set. Notation
or, in set-builder notation as
The union of set
The empty set, denoted by
Intervals are sets of real numbers. The open interval from
Absolute Value
The absolute value of a number
The distance between the points
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,
The number
Laws of Exponents
To multiply two powers of the same base, add the exponents:
To make
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
The exponent
The positive exponent
The negative exponent
Radicals
The symbol
Because
If
Rational Exponents
To give meaning to the symbol
So by definition of
In general, rational exponents are defined as follows: