What is Additive Inverse?
The additive inverse of a rational number \(a\) is the number that, when added to \(a\), gives zero (the additive identity). For any rational number \(\frac{p}{q}\), its additive inverse is \(\frac{-p}{q}\).
- Formula: \(a + (-a) = 0\)
- Example: Additive inverse of \(\frac{3}{5}\) is \(\frac{-3}{5}\) because \(\frac{3}{5} + \frac{-3}{5} = 0\)
What is Multiplicative Inverse (Reciprocal)?
The multiplicative inverse of a non-zero rational number \(a\) is the number that, when multiplied by \(a\), gives 1 (the multiplicative identity). For \(\frac{p}{q} \neq 0\), its multiplicative inverse is \(\frac{q}{p}\).
- Formula: \(a \times \frac{1}{a} = 1\) (where \(a \neq 0\))
- Example: Multiplicative inverse of \(\frac{2}{3}\) is \(\frac{3}{2}\) because \(\frac{2}{3} \times \frac{3}{2} = 1\)
Real-life Connection: Think of additive inverse like walking forward 5 steps and then backward 5 steps to return to start. Multiplicative inverse is like doubling a recipe then halving it to get back to original!
Representation on Number Line
Rational numbers can be plotted on a number line - a straight line with numbers placed at equal intervals. Zero is at the center, positive numbers to the right, negative numbers to the left.
Steps to represent a rational number on number line:
- Draw a horizontal line and mark 0 in the middle
- Mark equal divisions (based on the denominator)
- Locate the number based on numerator (positive = right, negative = left)