Numbers are mathematical objects used for counting and measuring.
The main classifications are:
- Natural Numbers \(\mathbb{N}\): \(1, 2, 3, 4, \dots\) (Counting numbers).
- Whole Numbers \(\mathbb{W}\): \(0, 1, 2, 3, 4, \dots\) (Natural numbers plus zero).
- Integers \(\mathbb{Z}\): \(\dots, -3, -2, -1, 0, 1, 2, 3, \dots\) (Whole numbers and their negatives).
- Rational Numbers \(\mathbb{Q}\): Numbers that can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
Important!
Every natural number is a whole number, an integer, and a rational number.
Every whole number is an integer and a rational number.
Every integer is a rational number.
Rational Numbers \(\mathbb{Q}\):
Equivalent Representation: A rational number does not have a unique representation in the form \(\frac{p}{q}\).
Example:
\(\frac{3}{4} = \frac{6}{8} = \frac{9}{12}\). These are called equivalent rational numbers.
Important!
There are infinitely many rational numbers between any two given rational numbers.
Finding Rational Numbers:
Two methods to find rational numbers between \(a\) and \(b\):Average Method: The average \(\frac{a+b}{2}\) is a rational number between \(a\) and \(b\). This process can be repeated infinitely.
Same Denominator Method: To find \(n\) rational numbers between \(a\) and \(b\), write them with a denominator of \(n+1\): \(a=\frac{a\times(n+1)}{n+1}\) and \(b=\frac{b\times(n+1)}{n+1}\)
Decimal Expansion:
The decimal expansion of a rational number is either terminating or non-terminating and recurring (periodic).
Irrational Numbers:
An irrational number is a number that cannot be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). i.e., \(\sqrt{2}, \sqrt{3}, \pi,...\)
Decimal Expansion: Irrational numbers have decimal expansions that are neither terminating nor recurring (non-terminating and non-periodic).
Important!
As there are infinitely many rationals, there are also infinitely many irrational numbers.
Real Numbers \(\mathbb{R}\):
Real numbers are the numbers that include both rational and irrational numbers. \(\mathbb{R}\) is the union of the set of rational and irrational numbers.
Number Line Correspondence (Cantor and Dedekind):
- Corresponding to every real number, there is a unique point on the real number line.
- Corresponding to every point on the number line, there exists a unique real number.
- This means the real number line is "filled" by rational and irrational numbers.
Now we will analyse the decimal expansions of different types of numbers.
|
Numbers
|
Nature of the decimal expansion
|
Type of number
|
| \(\frac{1}{125} = 0.008\) | Terminating expansion. | Rational |
| \(\frac{1}{6} = 0.1666... =\overline {0.16}\) | Recurring and non-terminating decimals. | Rational |
| \(\frac{152}{333} = 0.456456456... = \overline{0.456}\) | Recurring and non-terminating decimals. | Rational |
| \(π=3.14159265358979323846...\) | Non -repeating and non- terminating decimals. | Irrational |
Finding irrational numbers between two numbers:
Steps to find the irrational numbers between two numbers:
Step 1: Express the provided number in decimal form.
Step 2: Keep in mind that we need to construct irrational numbers between the provided numbers.
Step 3: Within the possible range construct irrational numbers such that they are non-recurring and non-terminating.
Step 2: Keep in mind that we need to construct irrational numbers between the provided numbers.
Step 3: Within the possible range construct irrational numbers such that they are non-recurring and non-terminating.