Number - Operations with Rational Numbers (Integers, Fractions, Decimals)

Rational Numbers Definition: A rational number is any value that can be written as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Visually, these numbers occupy specific points on a number line, including all integers, terminating decimals (like $0.75 = \frac{3}{4}$), and recurring decimals (like $0.333... = \frac{1}{3}$).

Integer Operations and Sign Rules: When operating with integers, the signs determine the direction of movement on a number line. For multiplication and division, like signs result in a positive value ($(-) \times (-) = (+)$), while unlike signs result in a negative value ($(+) \times (-) = (-)$). Visually, multiplying by a negative number acts as a $180^{\circ}$ rotation or 'flip' to the opposite side of zero.

Order of Operations (BIDMAS/BODMAS): Calculations must follow the sequence of Brackets, Indices, Division/Multiplication (left to right), and Addition/Subtraction (left to right). Visually, imagine a priority pyramid where Brackets sit at the narrow top and Addition/Subtraction form the wide base; work from the top down to ensure accuracy.

Adding and Subtracting Fractions: To add or subtract, fractions must have a Common Denominator. Visually, if you have a circle cut into thirds and another into quarters, you cannot combine the pieces until you re-divide both circles into $12$ equal parts (the Least Common Multiple of $3$ and $4$) so the 'slice sizes' are identical.

Multiplying and Dividing Fractions: Multiplying fractions involves multiplying the numerators together and the denominators together. For division, use the 'Keep-Change-Flip' rule by multiplying the first fraction by the reciprocal of the second. Visually, multiplying $\frac{1}{2} \times \frac{1}{3}$ is equivalent to finding 'one-third of one-half' of a rectangle, resulting in $\frac{1}{6}$ of the total area.

Decimal Place Value and Operations: When adding or subtracting decimals, align the decimal points vertically to ensure digits of the same place value (tenths, hundredths, etc.) are combined correctly. Visually, think of place value columns where the decimal point acts as a fixed anchor between the 'Ones' and 'Tenths' positions.

Absolute Value: The absolute value $|x|$ represents the distance of a number from zero on a number line, regardless of direction. Visually, both $-7$ and $7$ have an absolute value of $7$ because they are both exactly $7$ units away from the center point $0$.

Converting Between Forms: Rational numbers can be converted between fractions, decimals, and percentages. Visually, a $10 \times 10$ square grid representing 'one whole' can be used: shading $20$ squares represents $20\%$, $0.20$ as a decimal, and $\frac{20}{100}$ (or $\frac{1}{5}$) as a fraction.