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Cancelling out is amathematical process used for removing subexpressions from amathematical expression, when this removal does not change the meaning or the value of the expression because the subexpressions have equal and opposing effects.^[1] For example, afraction is put inlowest terms by cancelling out thecommon factors of thenumerator and thedenominator.^[2] As another example, ifa×b=a×c, then the multiplicative terma can be canceled outifa≠0, resulting in the equivalent expressionb=c; this is equivalent to dividing through bya.

If the subexpressions are not identical, then it may still be possible to cancel them out partly. For example, in the simple equation 3 + 2y = 8y, both sides actually contain 2y (because 8y is the same as 2y + 6y). Therefore, the 2y on both sides can be cancelled out, leaving 3 = 6y, ory= 0.5. This is equivalent to subtracting 2y from both sides.

At times, cancelling out can introduce limited changes or extra solutions to anequation. For example, given the inequalityab ≥ 3b, it looks like theb on both sides can be cancelled out to givea ≥ 3 as the solution. But cancelling 'naively' like this, will mean we don't get all the solutions (sets of (a, b) satisfying the inequality). This is because ifb were anegative number then dividing by a negative would change the ≥ relationship into a ≤ relationship. For example, although 2 is more than 1, –2 isless than –1. Also ifb werezero then zero times anything is zero and cancelling out would meandividing by zero in that case which cannot be done. So in fact, while cancelling works, cancelling out correctly will lead us tothree sets of solutions, not just one we thought we had. It will also tell us that our 'naive' solution is only a solution in some cases, not all cases:

• Ifb > 0: we can cancel out to geta ≥ 3.
• Ifb < 0: then cancelling out givesa ≤ 3 instead, because we would have to reverse the relationship in this case.
• Ifb is exactly zero: then the equation is true forany value ofa, because both sides would be zero, and 0 ≥ 0.

So some care may be needed to ensure that cancelling out is done correctly and no solutions are overlooked or incorrect. Our simple inequality hasthree sets of solutions, which are:

• b > 0 anda≥ 3. (For exampleb= 5 anda= 6 is a solution because 6 x 5 is 30 and 3 x 5 is 15, and 30 ≥ 15)
  or
• b< 0 anda≤ 3 (For exampleb= –5 anda= 2 is a solution because 2 x (–5) is –10 and 3 x (–5) is –15, and –10 ≥ –15)
  or
• b= 0 (anda can be any number) (becauseanythingx zero ≥ 3 x zero)

Our 'naïve' solution (thata≥ 3) would also be wrong sometimes. For example, ifb= –5 thena= 4 is not a solution even though 4 ≥ 3, because 4 × (–5) is –20, and 3 x (–5) is –15, and –20 is not ≥ –15.

In more advanced mathematics, cancelling out can be used in the context ofinfinite series, whose terms can be cancelled out to get a finite sum or aconvergent series. In this case, the termtelescoping is often used. Considerable care and prevention of errors is often necessary to ensure the amended equation will be valid, or to establish thebounds within which it will be valid, because of the nature of such series.

Incomputational science, cancelling out is often used for improving theaccuracy and theexecution time ofnumerical algorithms.

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