Number System Definitions

 Number System
1. Natural Numbers
2. Whole Numbers
3. Integers
4. Rational Numbers
5. Real Numbers
6. Irrational Numbers

Natural Numbers (ℕ): Natural numbers are those numbers which starts from 1.
Examples: 1, 2, 3, 4, 5, ..........

Whole Numbers (W): Whole

 numbers are those numbers which starts from 0.
Examples: 0, 1, 2, 3, 4, 5, ..........

Integers (ℤ): 

These include all whole numbers, both positive numbers and negative numbers, as well as zero.
Examples: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational Numbers (ℚ):

 

These are numbers that can be expressed as P/Q or the ratio of two integers, where the denominator is not zero.
Examples: 1/2, -3/4, 5, 0.75 (which is 3/4), etc.

1. Fractions:

   • $\frac{1}{2}$ (One-half)
   • $\frac{-3}{4}$ (Negative three-fourths)
   • $\frac{5}{3}$ (Five-thirds)
   • $\frac{7}{1}$ (Seven is an integer, but it can be written as a rational number)
   • $\frac{-2}{5}$ (Negative two-fifths)

2. Integers (because they can be written as a fraction with a denominator of 1):

   • 3 (can be written as $\frac{3}{1}$)
   • -6 (can be written as $\frac{-6}{1}$)
   • 0 (can be written as $\frac{0}{1}$)

3. Decimals (as long as the decimal is either terminating or repeating):

   • 0.5 (can be written as $\frac{1}{2}$)
   • 1.25 (can be written as $\frac{5}{4}$)
   • -0.333... (which repeats and can be written as $\frac{-1}{3}$)
   • 0.75 (can be written as $\frac{3}{4}$)
   • -2.5 (can be written as $\frac{-5}{2}$)

4. Repeating Decimals:

   • 0.666... (which repeats and can be written as $\frac{2}{3}$)
   • 1.272727... (which repeats and can be written as $\frac{14}{11}$)

Real Numbers (ℝ): 

These include both rational and irrational numbers. They represent all the numbers that can be found on the number line.
Examples: 5, -1/2, √2, π, 0.333...

1. Rational Numbers (numbers that can be expressed as fractions $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$):

• Integers (numbers that can be written as $\frac{a}{1}$):
  • 5, -3, 0, 10, -12
• Fractions:
  • $\frac{1}{2}$, $\frac{-3}{4}$, $\frac{7}{8}$, $\frac{5}{3}$
• Decimals (either terminating or repeating):
  • 0.5 (can be written as $\frac{1}{2}$)
  • 1.25 (can be written as $\frac{5}{4}$)
  • 0.333... (repeating, can be written as $\frac{1}{3}$)
  • 0.75 (can be written as $\frac{3}{4}$)
  • -2.5 (can be written as $\frac{-5}{2}$)

2. Irrational Numbers (numbers that cannot be expressed as fractions and have non-terminating, non-repeating decimals):

• Square roots of non-perfect squares:

  • $\sqrt{2}$ (approximately 1.414213562...)
  • $\sqrt{3}$ (approximately 1.732050807...)
  • $\sqrt{5}$ (approximately 2.236067977...)
  • $\sqrt{7}$ (approximately 2.645751311...)

• Pi (π) (the ratio of the circumference of a circle to its diameter):

  • $\pi$ (approximately 3.141592653...)

• Euler's number (e) (the base of the natural logarithm):

• $e$ (approximately 2.718281828...)

• Non-repeating, non-terminating decimals:

          1.101001000100001... (a number where the pattern of 0s and 1s never repeats)

3. Special Cases of Real Numbers:

• Zero (0) is a real number and can be considered both rational (as $\frac{0}{1}$) and irrational (since it fits within the category of real numbers).

4. Negative and Positive Real Numbers:

• Positive real numbers: 5, 3.14, $\sqrt{2}$, $\pi$, 1.5
• Negative real numbers: -2, -4.5, $-\sqrt{5}$, $-\pi$

Irrational Numbers: 

These are real numbers that cannot be written as a simple fraction (ratio of two integers). They have non-repeating, non-terminating decimal expansions.
Examples: √2, π, e, etc.

1. Square Roots of Non-Perfect Squares:

• $\sqrt{2}$ (approximately 1.414213562...)
• $\sqrt{3}$ (approximately 1.732050807...)
• $\sqrt{5}$ (approximately 2.236067977...)
• $\sqrt{7}$ (approximately 2.645751311...)
• $\sqrt{11}$ (approximately 3.31662479...)
• $\sqrt{13}$ (approximately 3.605551275...)

2. Pi (π):

• $\pi$ (the ratio of the circumference of a circle to its diameter) is approximately 3.14159265358979... and is non-terminating and non-repeating.

3. Non-Repeating, Non-Terminating Decimals:

• 1.101001000100001... (A number where the pattern of 0s and 1s never repeats)
• 0.101001000100001... (another example of a non-repeating, non-terminating decimal)

4. Other Famous Irrational Numbers:

• $\sqrt[3]{2}$ (cube root of 2)