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Negative Reciprocal

Negative Reciprocal
Opposite Reciprocal

The result of taking the reciprocal of a number and then changing the sign. For example, the negative reciprocal of 5 is negative one-fifth (−1/5), and the negative reciprocal of -2/3 is The fraction 3/2.

Note: Perpendicular lines have slopes that are negative reciprocals of each other.

Key Formula

Negative reciprocal of a=−1a\text{Negative reciprocal of } a = -\frac{1}{a}
Where:
  • aa = Any nonzero real number

Worked Example

Problem: Line 1 has a slope of 3/4. Find the slope of a line perpendicular to it.
Step 1: Take the reciprocal of the slope by flipping the fraction.
34→43\frac{3}{4} \rightarrow \frac{4}{3}
Step 2: Change the sign. The original slope is positive, so make it negative.
43→−43\frac{4}{3} \rightarrow -\frac{4}{3}
Step 3: Verify by multiplying the two slopes. If they are negative reciprocals, the product must be −1-1.
34×(−43)=−1212=−1✓\frac{3}{4} \times \left(-\frac{4}{3}\right) = -\frac{12}{12} = -1 \checkmark
Answer: The slope of the perpendicular line is −43-\dfrac{4}{3}.

Another Example

Problem: Find the negative reciprocal of −6-6.
Step 1: Write −6-6 as a fraction and take its reciprocal.
−6=−61→1−6-6 = \frac{-6}{1} \rightarrow \frac{1}{-6}
Step 2: Change the sign. The original number is negative, so the result becomes positive.
1−6→16\frac{1}{-6} \rightarrow \frac{1}{6}
Step 3: Check: multiply −6-6 by 16\frac{1}{6}.
−6×16=−1✓-6 \times \frac{1}{6} = -1 \checkmark
Answer: The negative reciprocal of −6-6 is 16\dfrac{1}{6}.

Frequently Asked Questions

What is the negative reciprocal of a negative number?
When you start with a negative number, the two sign changes (one from taking the reciprocal's original sign, one from flipping the sign) produce a positive result. For example, the negative reciprocal of −2-2 is 12\frac{1}{2}, because 1−2\frac{1}{-2} flipped in sign gives 12\frac{1}{2}. You can always check: −2×12=−1-2 \times \frac{1}{2} = -1.
Why do perpendicular lines have slopes that are negative reciprocals?
When two lines cross at a right angle, one line rises at the same rate the other runs, and vice versa — which is why the fractions flip. The opposite sign ensures one line goes uphill while the other goes downhill. Algebraically, this relationship guarantees that the product of their slopes equals −1-1, which is the precise condition for perpendicularity in coordinate geometry.

Reciprocal (Multiplicative Inverse) vs. Negative Reciprocal

The reciprocal of aa is 1a\frac{1}{a}; their product is 11. The negative reciprocal of aa is −1a-\frac{1}{a}; their product is −1-1. The reciprocal keeps the same sign, while the negative reciprocal always switches it. You use reciprocals when dividing fractions; you use negative reciprocals when working with perpendicular slopes.

Why It Matters

Negative reciprocals are the key to solving any problem involving perpendicular lines in coordinate geometry. Whenever you need to find the equation of a line perpendicular to a given line, you take the negative reciprocal of the known slope. This concept also appears in physics and engineering when dealing with orthogonal directions or normal vectors.

Common Mistakes

Mistake: Only flipping the fraction without changing the sign (or only changing the sign without flipping).
Correction: You must do both operations: flip the fraction AND switch the sign. For instance, the negative reciprocal of 25\frac{2}{5} is −52-\frac{5}{2}, not −25-\frac{2}{5} and not 52\frac{5}{2}.
Mistake: Thinking the negative reciprocal of a negative number is also negative.
Correction: The negative reciprocal of a negative number is positive. For example, the negative reciprocal of −3-3 is 13\frac{1}{3} (positive), because the sign flips from negative to positive.

Related Terms