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The difference of two squares can also be used in the rationalising of irrational denominators. [2] This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots .
Subtraction is non-associative, which comes up when one tries to define repeated subtraction. In general, the expression "a − b − c" can be defined to mean either (a − b) − c or a − (b − c), but these two possibilities lead to different answers.
The lowest common denominator of a set of fractions is the lowest number that is a multiple of all the denominators: their lowest common multiple. The product of the denominators is always a common denominator, as in: + = + =
The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the ...
If a and b have different signs, define a + b to be the difference between |a| and |b|, with the sign of the term whose absolute value is larger. [61] As an example, −6 + 4 = −2; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative.
The smallest common multiple of the two denominators 6 and 15z is 30z, so one multiplies both sides by 30z: + =. The result is an equation with no fractions. The simplified equation is not entirely equivalent to the original.
This is a simple example of double counting, often used when teaching multiplication to young children. In this context, multiplication of natural numbers is introduced as repeated addition, and is then shown to be commutative by counting, in two different ways, a number of items arranged in a rectangular grid.
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