Understanding Rational Numbers
A rational number is any number that can be expressed as a fraction a⁄b where a and b are integers and b ≠ 0. Both positive and negative integers are allowed, but the denominator must never be zero.
Adding and Subtracting Rational Numbers
Before you can add or subtract fractions, you must find a common denominator. Only fractions with the same denominator can be combined directly.
When the signs of the addends are the same, add the absolute values and keep the common sign. When the signs differ, subtract the smaller absolute value from the larger one and adopt the sign of the larger absolute value.
Example: 3/8 = 3 ÷ 8 = 0.375. Dividing the numerator by the denominator gives the decimal equivalent.
Special Cases with Negative Numbers
- Among negative numbers, the one with the greater absolute value is smaller on the number line.
- Subtracting a negative number is the same as adding its positive counterpart: a - (-b) = a + b.
Sign Rules for Multiplication and Division
Multiplying two numbers with the same sign always yields a positive product. The same rule applies to division.
Summary of sign rules:
| Operation | Same Sign | Different Sign |
|---|---|---|
| Multiplication | + | - |
| Division | + | - |
Working with Percentages
Before multiplying a percent by a number, convert the percent to a decimal by dividing by 100 (move the decimal point two places left). For example, 20% becomes 0.20.
To find a discount, multiply the original price by the decimal form of the percent. Example: An item costs ₱600 with a 20% discount. ₱600 × 0.20 = ₱120 discount.
Converting Fractions to Percents
Follow a two‑step process:
- Convert the fraction to a decimal (divide numerator by denominator).
- Convert the decimal to a percent (multiply by 100).
Perfect Squares and Square Roots
A perfect square is the product of an integer multiplied by itself (n² = n × n). The principal square root of a perfect square is the non‑negative root. For example, √49 = 7, so the principal square root of 49 is 7.
When determining whether a square root is rational or irrational:
- If the radicand (the number under the root) is a perfect square, the root is rational.
- If the radicand is not a perfect square, the root is irrational.
Estimating Irrational Roots
To estimate an irrational square root, locate the number between the nearest perfect squares. For an irrational cube root, locate it between the nearest perfect cubes.
Place an irrational number on the number line between the two consecutive integers it falls between after estimating its value.
Perfect Cubes and Cube Roots
A perfect cube is an integer multiplied by itself three times (n³ = n × n × n). The cube root of a number is the value that, when cubed, returns the original number. Negative numbers have real cube roots; for example, the cube root of –216 is –6 because (–6)³ = –216.
Comparing Decimals and Ordering Numbers
When comparing decimals, add trailing zeros to align place values: 0.5 = 0.500.
On a number line, numbers farther to the right are greater. This principle also applies to negative numbers: a more negative value lies farther left and is therefore smaller.
Key Takeaways
- Rational numbers are fractions with non‑zero denominators.
- Common denominators are required for fraction addition/subtraction.
- Sign rules for multiplication and division are identical.
- Convert percents to decimals before using them in calculations.
- Perfect squares and cubes have integer roots; non‑perfect radicands produce irrational results.
- Estimate irrational roots by locating them between nearest perfect powers.
- Use trailing zeros to compare decimals accurately.
