Algebra

Basics of Real Numbers

Big Picture

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.

Key Terms

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 \(\Pi\)

Real numbers overview chart

Symbols

Here are some common symbols used in algebra:

Symbols

Symbol
Meaning
+
Add
-
Substract
\(\times\) or \(\cdot\)
Multiply
\(\div\) or /
Divide
\(\sqrt{\phantom{x}}\) or \(\sqrt[n]{\phantom{x}}\)
square root, nth root
| |
Absolute value
=
Equals
Not equal
Approximately equal
<, ≤
Less than, less than or equal to
>, ≥
Greater than, greater than or equal to
{  }
Set symbol
\(\in\)
An element of a set
( ), [ ]
Group symbols
Symbol
Meaning
+
Add
-
Substract
X or .
Multiply
\(\div\) or /
Divide
\(\sqrt{}\) or \(\sqrt[n]{}\)
square root, nth root
| |
Absolute value
=
Equals
Not equal
Approximately equal
<, ≤
Less than, less than or equal to
>, ≥
Greater than, greater than or equal to
{  }
Set symbol
\(\in\)
An element of a set
( ), [ ]
Group symbols

Understanding Real Numbers

  • Sum or product of two rational numbers is rational.
  • Example: \(2+3=5\)
  • Example: \(\frac{2}{3} . \frac{4}{5}\) =  \(\frac{8}{15}\)
  • Sum of rational number and irrational number is irrational
  • Example: \(2 + \sqrt{2} = 2 + \sqrt{2}\)
  • Product of nonzero rational number and irrational number is irrational.
  • Example: \(3 . \sqrt{2} = 3\sqrt{2}\)
  • Example: \(5 . \pi = 5 \pi\)
  • Difference between two whole rational numbers is not always a positive number.
  • Example: \(5 - 4 = 1\)
  • Example: \(5 - 9 = -4\)
  • Quotient of a whole rational divisor and a whole  dividend is not always a whole number.
  • Example: \(\frac{6}{3} = 2\)
  • Example: \(\frac{4}{8} = \frac{1}{2}\)

Algebra

Basics of Real Numbers Cont.

Exponents and nth Roots

Exponents

Exponent is a short-hand notation for repeated multi-plication.

  • \(2 · 2 · 2 = 2^{3}\). We say that 2 is raised to the power of 3.
  • \(2 · 2 · 2 · 2 · 2 = 2\sqrt{5}\) . We say that 2 is raised to the power of 5.
  • For \(x^n\) , we say that x is raised to the power of n.
  • x and n are variables, symbols that are used to represent a value.
  • If  n=2, we can also say x squared. If x=3, we say x cubed.

nth Roots

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 \(\sqrt[n]{y} = x\)

  • \(\sqrt{16} = 4\) because \(4^{2} = 16\)
  • \(\sqrt{\phantom{x}}\) is called the radical sign
  • When n=2, we usually write \(\sqrt{\phantom{x}}\) , not \(\sqrt[2]{\phantom{x}}\), and we call it the square root.
  • When n=3, we call it the cube root.

If the nth root can’t be simplified (reduced) to a rational number without the radical sign (\(\sqrt{\phantom{x}}\) ), the number is irrational.

  • Examples: \(\sqrt{64}  = 8\), so it is a rational number. \(\sqrt{2}\) cannot be reduced any further and 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.

  • Example: \(\sqrt{2} ≈ 1.414\) when rounded to three decimal places.

Fractions and Decimals

A rational number is just a ratio of one number to another written in fraction form as \(\frac{a}{b}\)

  • Every whole number is a rational number where the denominator equals 1.
  • A denominator equal to 1 is sometimes called the invisible denominator because it is not usually written out: \(\frac{a}{1} = a\)
  • Fraction bar: the line that separates the numerator and the denominator. The denominator b ≠ 0.
  • A proper fraction represents a number less than one because a < b, while an improper fraction represents a number greater than one because a > b.
  • A negative fraction is usually written with the negative sign to the left of the fraction.
  • Example: \(-\frac{1}{2}\) could equal \(\frac{-1}{2}\) or \(\frac{1}{-2}\)
  • Improper fractions can be rewritten as an integer plus a proper fraction (mixed number).
  • Example: \(\frac{14}{3}\) = \(4\frac{2}{3}\)
  • Whenever we can write two fractions equal (=) to each other, we have equivalent fractions.
  • Example: \(\frac{2}{4}=\frac{1}{2}\), so \(\frac{2}{4}\) and  \(\frac{1}{2}\) are equivalent fractions.
  • The fractions \(\frac{a}{b} = \frac{c. a}{c . b}\) are equivalent as long as c ≠ 0.
Figure: Equivalent fractions

A fraction can be converted into a decimal - just divide the numerator by the denominator.

  • Example: \(\frac{1}{2}=0.5, \frac{1}{3}=0.333...\)
  • The ... means that the decimal goes on forever.

Not all decimals can be converted into fractions.

  • If the numbers after the decimal point (.) never repeats and never ends, the number is irrational.
  • Any number that can’t be written as a fraction is irrational.

Algebra

Basics of Real Numbers Cont.

Sets

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.

  • Numbers like 2.5, -3, and 7 are not elements of A.
  • We can also write that 1 \(\in\) A, meaning the number 1 is an element in set A.
  • If there are no elements in the set, we call it a null set or an empty set.

Union: the set of all elements that belong to A or B

  • Denoted as \(A \bigcup B\)
A union B
  • The union of rational numbers and irrational numbers is all real numbers.

Intersection: the set of elements that is true for both A and B

  • Denoted as \(A \bigcap B\)
A union B

Difference: the set of elements that belong to A only

  • Denoted as A \ B
A union B
  • If A is the group of whole numbers and B is the group of natural numbers, A \ B is 0

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

  • Denoted as \(A^{c}\)
A union B
  • B \ A is equal to \(A^{c}\). If A is the group of whole numbers and B is the group of natural numbers, \(A^{c}\) is null (there are no elements in set B that is not also in A)

Disjoint Sets: when sets A and B have no common elements.

  • Rational and irrational numbers are disjoint sets.
A union B

Notes