1.1 Integers
Integers are whole numbers with positive or negative signs, including zero. They do not contain fractions or decimals.
- Positive Integers: Numbers greater than zero (e.g., +1, +2, +15 or simply 1, 2, 15).
- Negative Integers: Numbers less than zero, denoted with a negative sign (e.g., -1, -5, -20).
- Zero (0): Neither positive nor negative.
Representation on a Number Line
On a horizontal number line, values increase to the right and decrease to the left.
1.2 Basic Arithmetic Operations Involving Integers
Rules for Combining Signs:
- +(+) = + | -(-) = + (Same signs yield positive)
- +(-) = - | -(+) = - (Different signs yield negative)
Multiplication and Division of Integers:
- (+) × (+) = (+) | (-) × (-) = (+)
- (+) × (-) = (-) | (-) × (+) = (-)
Laws of Arithmetic Operations:
- Commutative Law: $a + b = b + a$ | $a \times b = b \times a$
- Associative Law: $(a + b) + c = a + (b + c)$ | $(a \times b) \times c = a \times (b \times c)$
- Distributive Law: $a \times (b + c) = a \times b + a \times c$ | $a \times (b - c) = a \times b - a \times c$
- Identity Law: $a + 0 = a$ | $a \times 1 = a$ | $a \times 0 = 0$ | $a + (-a) = 0$
1.3 Positive and Negative Fractions
Fractions can be positive or negative. A negative fraction lies to the left of zero on the number line.
- Comparing Fractions: Express fractions with a common denominator to compare or order them.
- Operations: Follow the standard order of operations (BODMAS) when performing calculations with positive and negative fractions.
1.4 Positive and Negative Decimals
Decimals on the left side of zero are negative decimals, while those on the right are positive decimals.
- Ordering Decimals: Align decimals relative to zero on the number line to compare magnitude.
- Calculations: Perform calculations strictly using parentheses first, followed by multiplication/division from left to right, then addition/subtraction.
1.5 Rational Numbers
A Rational Number is any number that can be expressed in the fraction form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
Examples:
- Integers: $5 = \frac{5}{1}$ (Rational)
- Terminating Decimals: $0.75 = \frac{3}{4}$ (Rational)
- Repeating Decimals: $0.333... = \frac{1}{3}$ (Rational)
- Mixed Numbers: $1 \frac{1}{2} = \frac{3}{2}$ (Rational)