The Ultimate Guide to Numbers

1. Types of Numbers

In mathematics, we categorize numbers based on their properties:

Integers (\(\mathbb{Z}\)): Whole numbers, including negative, positive, and zero.
Examples: \(-3, 0, 42\)
Positive & Negative: Positive numbers are \(> 0\). Negative numbers are \(< 0\). Adding a negative is the same as subtracting a positive.
Zero: Neither positive nor negative. It acts as the "placeholder" and the identity element for addition.
Fractions & Decimals: Ways to represent parts of a whole.

2. Place Value

The position of a digit determines its value. For the number 4,321.56:

Thousands	Hundreds	Tens	Units	.	Tenths	Hundredths
4 (4000)	3 (300)	2 (20)	1 (1)	.	5 (\(\frac{5}{10}\))	6 (\(\frac{6}{100}\))

3. Symbols of Comparison

\(=\) Equals: Both sides are identical in value.
\(\neq\) Not Equals: The sides are different.
\(>\) Greater Than: The left side is larger. \(10 > 7\)
\(<\) Less Than: The left side is smaller. \(-5 < 2\)

4. Fractions: Proper, Improper, and Mixed

Proper Fraction: Numerator is smaller than denominator. \(\frac{3}{4}\)
Improper Fraction: Numerator is larger. \(\frac{7}{4}\)
Mixed Number: A whole number and a fraction. \(1\frac{3}{4}\)

Conversion Example: To change \(2\frac{1}{3}\) to an improper fraction:
Multiply whole number by denominator, add numerator: \((2 \times 3) + 1 = 7\). Result: \(\frac{7}{3}\).

5. Fractions to Decimals (and back)

Fraction \(\rightarrow\) Decimal: Divide the top by the bottom.
\(\frac{5}{8} = 5 \div 8 = 0.625\) (Terminating)
\(\frac{1}{3} = 1 \div 3 = 0.333...\) (Recurring, written as \(0.\dot{3}\))

Decimal \(\rightarrow\) Fraction: Use place value.
\(0.75 = \frac{75}{100} = \frac{3}{4}\)

6. Rational vs Irrational Numbers

Type	Definition	Examples
Rational	Can be written as a fraction \(\frac{p}{q}\) where \(p, q\) are integers.	\(5, -0.2, \frac{2}{3}, 0.\dot{1}\)
Irrational	Cannot be written as a fraction. Decimals go on forever without repeating a pattern.	\(\pi, \sqrt{2}, \sqrt{3}\)