Calculator
Enter a number, fraction, decimal, or mixed number to find its opposite reciprocal.
Example data table
These sample values show how the calculator handles integers, fractions, decimals, and mixed numbers.
| Input | Simplified value | Reciprocal | Opposite reciprocal | Use case |
|---|---|---|---|---|
| 2 | 2 | 1/2 | -1/2 | Perpendicular slope to 2 |
| -3/4 | -3/4 | -4/3 | 4/3 | Signed fraction example |
| 1.25 | 5/4 | 4/5 | -4/5 | Decimal converted exactly |
| 2 1/2 | 5/2 | 2/5 | -2/5 | Mixed number input |
| -0.2 | -1/5 | -5 | 5 | Negative decimal example |
Formula used
If a value is written as x = a/b and x ≠ 0:
Reciprocal: 1/x = b/a
Opposite reciprocal: -(1/x) = -b/a
Slope connection: If a line has slope m, then a perpendicular line has slope -1/m.
How to use this calculator
- Enter a value as an integer, decimal, fraction, or mixed number.
- Choose the number of decimal places for rounded output.
- Select the graph range to control the plotted x-axis width.
- Click Calculate Opposite Reciprocal.
- Review the result cards, summary table, and step-by-step working.
- Use the CSV or PDF buttons to export your output.
- Check the Plotly graph to compare the original and perpendicular slopes.
FAQs
1) What is an opposite reciprocal?
An opposite reciprocal is the negative of a number’s reciprocal. For a nonzero value x, the opposite reciprocal is -1/x. It is also called the negative reciprocal in algebra and coordinate geometry.
2) Why is the opposite reciprocal useful?
It is mainly used for perpendicular slopes. If one line has slope m, a perpendicular line has slope -1/m, provided m is not zero or undefined. This helps in graphing and analytic geometry problems.
3) How do I find the opposite reciprocal of a fraction?
Flip the fraction to get the reciprocal, then change the sign. For example, 3/5 becomes 5/3, and its opposite reciprocal becomes -5/3. If the original fraction is negative, the opposite reciprocal becomes positive.
4) Can I enter mixed numbers?
Yes. Enter mixed numbers like 2 1/3 or -4 1/2. The calculator converts them into improper fractions first, simplifies them, then computes the reciprocal and the opposite reciprocal.
5) What happens if I enter zero?
Zero has no reciprocal because division by zero is undefined. Since the opposite reciprocal depends on the reciprocal, the opposite reciprocal of zero is also undefined.
6) Does the calculator simplify fractions automatically?
Yes. Every valid fraction, mixed number, or decimal input is reduced to simplest terms before the result is displayed. That makes the exact answer easier to read and verify.
7) Why does the product check equal -1?
For any nonzero value x, multiplying x by its opposite reciprocal gives x × (-1/x) = -1. This confirms the sign change and reciprocal step were both done correctly.
8) Is the graph showing the number itself?
The graph shows two lines through the origin. One uses the original value as slope, and the other uses its opposite reciprocal. This makes the perpendicular relationship easier to visualize.