Natural and Integer Numbers
Foundations of number systems and algebraic operations
Natural Numbers
The set of natural numbers consists of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, ..., n+1. This set is denoted by the symbol ℕ.
Natural numbers are the numbers used for counting. This set is infinitely large because every number n has a successor n + 1. Natural numbers are fundamental in mathematics and are the basis for all other number systems.
Operations with Natural Numbers
For natural numbers, the following operations are defined:
\(\displaystyle a + b = c\)
Example: \(\displaystyle 5 + 3 = 8\)
In addition, two numbers (a + b) are combined to form a new number. Here, a and b are called addends and the result is the sum.
Examples of Addition
- \(\displaystyle 2 + 3 = 5\)
- \(\displaystyle 10 + 7 = 17\)
- \(\displaystyle 100 + 50 = 150\)
\(\displaystyle a \cdot b = c\)
Example: \(\displaystyle 5 \cdot 3 = 15\)
In multiplication, two numbers (a · b) are combined to form a new number. Here, a and b are called factors and the result is the product.
Examples of Multiplication
- \(\displaystyle 2 \cdot 3 = 6\)
- \(\displaystyle 5 \cdot 4 = 20\)
- \(\displaystyle 10 \cdot 10 = 100\)
Addition and multiplication are closed on the natural numbers. This means: when you add or multiply any two natural numbers, the result is always a natural number.
Integer Numbers
The integers extend the range of natural numbers to include the negative range. This set is denoted by the symbol ℤ.
The integers are: ..., -3, -2, -1, 0, 1, 2, 3, ... They contain all natural numbers, zero, and all negative numbers. This set allows us to work with subtraction without restrictions, unlike the natural numbers.
Why Do We Need Integer Numbers?
Natural numbers can be added and multiplied without restriction. However, subtraction and division have limitations:
Works with Natural Numbers
- \(\displaystyle 7 - 5 = 2\) ✓
- \(\displaystyle 8 \div 2 = 4\) ✓
- \(\displaystyle 10 - 3 = 7\) ✓
However, with \(\displaystyle 5 - 7\), we leave the range of natural numbers. The result \(\displaystyle -2\) falls into the range of integers.
\(\displaystyle 5 - 7 = -2\) (Result is an integer)
\(\displaystyle 3 - 8 = -5\) (Another example)
Comparison: Natural vs. Integer Numbers
| Property | Natural Numbers (ℕ) | Integer Numbers (ℤ) |
|---|---|---|
| Symbol | ℕ | ℤ |
| Range | 0, 1, 2, 3, ... | ..., -2, -1, 0, 1, 2, ... |
| Negative Numbers | No | Yes |
| Addition | Closed | Closed |
| Subtraction | Not always possible | Always possible |
| Multiplication | Closed | Closed |
| Division | Not always possible | Not always possible |
Integers are a superset of natural numbers. This means: all natural numbers are also integers, but not all integers are natural numbers.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction of two integers. This set is denoted by the symbol ℚ.
A rational number is any number that can be written as \(\displaystyle \frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). Rational numbers include all integers, fractions, and terminating or repeating decimals.
Examples of Rational Numbers
- \(\displaystyle \frac{1}{2} = 0.5\) (terminating decimal)
- \(\displaystyle \frac{1}{3} = 0.\overline{3}\) (repeating decimal)
- \(\displaystyle \frac{22}{7} \approx 3.142857...\)
- \(\displaystyle 5 = \frac{5}{1}\) (integers are rational)
- \(\displaystyle -\frac{3}{4} = -0.75\)
Rational numbers allow us to divide without leaving the set. However, not all decimal numbers are rational. For example, \(\pi\) and \(\sqrt{2}\) cannot be expressed as fractions.
Real Numbers
Real numbers include all rational and irrational numbers. This is the most comprehensive number system commonly used in mathematics. This set is denoted by ℝ.
Real numbers are all numbers that can be plotted on a number line. They include integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\).
\(\displaystyle \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)
Natural Numbers ⊂ Integers ⊂ Rationals ⊂ Real Numbers
Examples of Real Numbers
- \(\displaystyle 5\) (integer)
- \(\displaystyle 0.75 = \frac{3}{4}\) (rational)
- \(\displaystyle \pi \approx 3.14159...\) (irrational)
- \(\displaystyle \sqrt{2} \approx 1.41421...\) (irrational)
- \(\displaystyle e \approx 2.71828...\) (irrational)
- \(\displaystyle -\sqrt{5}\) (negative irrational)
Irrational numbers are real numbers that cannot be expressed as fractions of integers. Their decimal representations are non-terminating and non-repeating. Common examples include \(\pi\), \(e\), and \(\sqrt{n}\) where n is not a perfect square.
Summary: Number Systems
| Number Set | Symbol | Description | Examples |
|---|---|---|---|
| Natural | ℕ | Counting numbers starting from 0 | 0, 1, 2, 3, 4, ... |
| Integer | ℤ | Natural numbers + negative numbers | ..., -2, -1, 0, 1, 2, ... |
| Rational | ℚ | Fractions and decimals a/b (b≠0) | \(\frac{1}{2}, \frac{3}{4}, 0.5, -2.75\) |
| Real | ℝ | All rational and irrational numbers | \(\pi, e, \sqrt{2}, 5, 0.333...\) |
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