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For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. [2] As another example, if a×b=a×c, then the multiplicative term a can be canceled out if a≠0, resulting in the equivalent expression b=c; this is equivalent to dividing through by a.
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
As this example shows, when like terms exist in an expression, they may be combined by adding or subtracting (whatever the expression indicates) the coefficients, and maintaining the common factor of both terms. Such combination is called combining like terms or collecting like terms, and it is an important tool used for solving equations.
One can simplify the classification of monomial orders by assuming that the indeterminates are named x 1, x 2, x 3, ... in decreasing order for the monomial order considered, so that always x 1 > x 2 > x 3 > .... (If there should be infinitely many indeterminates, this convention is incompatible with the condition of being a well ordering, and ...
Using these rules, we can show that 5 / 10 = 1 / 2 = 10 / 20 = 50 / 100 , for example. As another example, since the greatest common divisor of 63 and 462 is 21, the fraction 63 / 462 can be reduced to lowest terms by dividing the numerator and denominator by 21:
Since we know that =, if we divide both sides of the equation by , we divide both sides of the equation by zero. This operation is undefined in arithmetic, and therefore deductions based on division by zero can be contradictory.
Then, if one person moves out and the other stays, they can pay back their half. Some people subtract an agreed-upon “usage charge” for the years they lived together.
We can reduce the fractions to lowest terms by noting that the two occurrences of b on the left-hand side cancel, as do the two occurrences of d on the right-hand side, leaving =, and we can divide both sides of the equation by any of the elements—in this case we will use d —getting =. Another justification of cross-multiplication is as ...