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For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands.
Cancelling 0 from both sides yields =, a false statement. The fallacy here arises from the assumption that it is legitimate to cancel 0 like any other number, whereas, in fact, doing so is a form of division by 0. Using algebra, it is possible to disguise a division by zero [17] to obtain an invalid proof. For example: [18]
A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in 7 + 8 = 15, but 15:00 reads as 3:00 on the clock face because clocks "wrap around" every 12 hours and the hour number starts again at ...
Another method is multiplication by 3. A number of the form 10x + y has the same remainder when divided by 7 as 3x + y. One must multiply the leftmost digit of the original number by 3, add the next digit, take the remainder when divided by 7, and continue from the beginning: multiply by 3, add the next digit, etc.
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.
For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as 20 / 5 = 4 , or 20 / 5 = 4 . [ 2 ] In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.
For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2 −23 or about 10 −7 in single precision, and exactly 2 −53 or about 10 −16 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.
E.g.: x**2 + 3*x + 5 will be represented as [1, 3, 5] """ out = list (dividend) # Copy the dividend normalizer = divisor [0] for i in range (len (dividend)-len (divisor) + 1): # For general polynomial division (when polynomials are non-monic), # we need to normalize by dividing the coefficient with the divisor's first coefficient out [i ...