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Reciprocal In Algebra

Turn it upside down!

Reciprocal of a Number

To get the reciprocal of a number, we divide 1 by the number:

Equation showing the reciprocal of 8 equals 1 divided by 8

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 .

Dog standing upside down on its head In other words turn it upside down

But no reciprocal of zero

We can't divide by zero (10 is undefined).

So when we say the reciprocal of x is 1x, we also mean that x cannot be zero.

Example: What is the Reciprocal of x/(x−1) ?

Answer: take x(x−1) and flip it upside down:

(x−1)x

(but x not equal to 0) ../numbers/images/reciprocal.js?numer=x&denom=(x-1)

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!

Diagram showing x/y flipping to y/x and then flipping back to x/y

Example:

The reciprocal of axy is yax

The reciprocal of yax is axy (back again)

Example: What is:

11/w

Answer: w

Why? because the reciprocal of 1/w is w/1 which is just w

Or with numbers: What is 1½ ? Divide 1 into halves, and the answer is 2

Notation

The reciprocal of "x" can be shown as:

1x or x-1

(see exponents)

311, 1621,308, 1622, 2313, 3153, 2311, 2312, 3151, 3152 Reciprocal Reciprocal of a Fraction

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