Rational Numbers Properties

Study Notes

Commutative Property: The order of the rational numbers does not change the result.
- a/b + c/d = c/d + a/b
- a/b × c/d = c/d × a/b
Associative Property: The order in which rational numbers are grouped does not change the result.
- (a/b + c/d) + e/f = a/b + (c/d + e/f)
- (a/b × c/d) × e/f = a/b × (c/d × e/f)
Distributive Property: Rational numbers can be distributed over addition.
- a/b × (c/d + e/f) = (a/b × c/d) + (a/b × e/f)
Additive Identity: The additive identity is 0, which does not change the value of a rational number when added to it.
- a/b + 0 = a/b
Additive Inverse: Each rational number has an additive inverse, which when added to the rational number, results in 0.
- a/b + (-a/b) = 0
Multiplicative Identity: The multiplicative identity is 1, which does not change the value of a rational number when multiplied by it.
- a/b × 1 = a/b
Multiplicative Inverse: Each non-zero rational number has a multiplicative inverse, which when multiplied by the rational number, results in 1.
- a/b × 1/(a/b) = 1

To add rational numbers, follow these steps:
1. Find the least common multiple (LCM) of the denominators.
2. Convert both rational numbers to have the LCM as the denominator.
3. Add the numerators and keep the denominator the same.

To subtract rational numbers, follow these steps:
1. Find the least common multiple (LCM) of the denominators.
2. Convert both rational numbers to have the LCM as the denominator.
3. Subtract the numerators and keep the denominator the same.

To multiply rational numbers, follow these steps:
1. Multiply the numerators.
2. Multiply the denominators.
3. Simplify the resulting rational number.

To divide rational numbers, follow these steps:
1. Invert the second rational number (i.e., flip the numerator and denominator).
2. Multiply the first rational number by the inverted second rational number.
3. Simplify the resulting rational number.

Note: These properties and operations can be applied to all rational numbers, and are essential in working with fractions, decimals, and percentages.

Rational numbers follow the commutative property, meaning the order of numbers does not change the result of addition or multiplication.
Associative property applies to rational numbers, where the order of grouping does not change the result of addition or multiplication.
Rational numbers follow the distributive property, where a number can be distributed over addition.
The additive identity is 0, which does not change the value of a rational number when added to it.
Each rational number has an additive inverse, which when added to the rational number, results in 0.
The multiplicative identity is 1, which does not change the value of a rational number when multiplied by it.
Each non-zero rational number has a multiplicative inverse, which when multiplied by the rational number, results in 1.

To add rational numbers, find the least common multiple (LCM) of the denominators.
Convert both rational numbers to have the LCM as the denominator.
Add the numerators and keep the denominator the same.

To subtract rational numbers, find the least common multiple (LCM) of the denominators.
Convert both rational numbers to have the LCM as the denominator.
Subtract the numerators and keep the denominator the same.

To divide rational numbers, invert the second rational number (i.e., flip the numerator and denominator).
Multiply the first rational number by the inverted second rational number.
Simplify the resulting rational number.