Reciprocal In Algebra
Turn it upside down!
Reciprocal of a Number
To get the reciprocal of a number, we divide 1 by the number:
Examples:
| Number | Reciprocal | As a Decimal |
|---|---|---|
| 2 | 12 | = 0.5 |
| 8 | 18 | = 0.125 |
| 1,000 | 11,000 | = 0.001 |
Reciprocal of a Variable
Likewise, the reciprocal of a variable x is 1x .
And the reciprocal of something more complicated like xy is yx .
In other words turn it upside down
But no reciprocal of zero
We can't divide by zero, as 10 is undefined.
So when we say the reciprocal of x is 1x, we also mean that x can't be zero.
Example: What's the Reciprocal of x/(x−1) ?
Answer: take x(x−1) and flip it upside down:
(x−1)x
More Examples:
| Expression | Reciprocal |
|---|---|
| 2x | 12x |
| 3x | x3 |
| (2x−3)(x+5) | (x+5)(2x−3) |
Flipping a Flip
When we take the reciprocal of a reciprocal we end up back where we started!
Example:
The reciprocal of axy is yax
The reciprocal of yax is axy (back again)
Example:
11/w
Answer: w
Why? because the reciprocal of 1w is w1 which is just w
Or with numbers: What's 1½ ? Divide 1 into halves, and the answer is 2
Multiplying Gives 1
A fun and useful rule: when we multiply an expression by its reciprocal, we get 1
xy × yx = xyxy = 1
This is why the reciprocal is also called the multiplicative inverse.
Notation
The reciprocal of "x" can be shown as:
1x or x-1
(see exponents)