Real numbers are used to measure the quantity of things in life. Almost any number you can think of is most likely to be a real number. Real numbers can be broken down into different types of numbers such as rational and irrational numbers. They can be visualized using number lines and operated on using set symbols and operators. General guidelines and rules are created to work with real numbers.
Rational Number: Ratio of one integer to another: \(\frac{numerator}{denominator}\), as long as the denominator is not equal to 0.
Integer : A rational number where the denominator is equal to 1. Includes natural numbers, negative natural numbers, and 0.
Natural Numbers: Counting numbers such as 1, 2, 3.
Whole Numbers: All natural numbers and 0.
Non-Integer: A rational number where the denominator is not equal to 1.
Proper Fraction: Numerator is less than denominator. Represents a number less than one.
Improper Fraction: Numerator is greater than denominator. Represents a number greater than 1.
Equivalent Fractions: Two fractions that represent the same amount.
Irrational Number: Number that cannot be expressed as a fraction, such as \( \sqrt{2} \) or \(\Pi\)
Here are some common symbols used in algebra:
Exponent is a short-hand notation for repeated multi-plication.
The nth root is the inverse operation of raising a number to the nth power. So the inverse operation of \(x\sqrt{n} = y\) is \(\sqrt[n]{y} = x\)
If the nth root can’t be simplified (reduced) to a rational number without the radical sign (\(\sqrt{\phantom{x}}\) ), the number is irrational.
We can get an approximate value for irrational square roots by using the calculator. In decimal form, the number will look like an unending string of numbers.
A rational number is just a ratio of one number to another written in fraction form as \(\frac{a}{b}\)
A fraction can be converted into a decimal - just divide the numerator by the denominator.
Not all decimals can be converted into fractions.
Sets are used to define groups of elements. In math, sets can be used to define different types of numbers, such as even and odd numbers. Outside of math, sets can also be used for other elements such as sets of keys or sets of clothing. The different types of sets (as shown below) are used to classify the objects in the sets.
We can list the elements (members) of a set inside the symbols { }. If A = {1, 2, 3}, then the numbers 1, 2, and 3 are elements of set A.
Union: the set of all elements that belong to A or B
Intersection: the set of elements that is true for both A and B
Difference: the set of elements that belong to A only
The order here matters! B \ A means the set of elements belonging to B only.
Complement Set: all elements in a set that is not A
Disjoint Sets: when sets A and B have no common elements.