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In general, if an increase of x percent is followed by a decrease of x percent, and the initial amount was p, the final amount is p (1 + 0.01 x)(1 − 0.01 x) = p (1 − (0.01 x) 2); hence the net change is an overall decrease by x percent of x percent (the square of the original percent change when expressed as a decimal number).
As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers. Thus, the ratio 40:60 is equivalent in meaning to the ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3.
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
Around 500 BC, the Greek mathematicians led by Pythagoras also realized that the square root of 2 is irrational. For Greek mathematicians, numbers were only the natural numbers. Real numbers were called "proportions", being the ratios of two lengths, or equivalently being measures of a length in terms of another length, called unit length.
In decimal numbers greater than 1 (such as 3.75), the fractional part of the number is expressed by the digits to the right of the separator (with a value of 0.75 in this case). 3.75 can be written either as an improper fraction, 375 / 100 , or as a mixed number, 3 + 75 / 100 .
In a fractional binary number such as 0.11010110101 2, the first digit is , the second () =, etc. So if there is a 1 in the first place after the decimal, then the number is at least , and vice versa. Double that number is at least 1.
1/52! chance of a specific shuffle Mathematics: The chances of shuffling a standard 52-card deck in any specific order is around 1.24 × 10 −68 (or exactly 1 ⁄ 52!) [4] Computing: The number 1.4 × 10 −45 is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.
These numbers also give the solution to certain enumerative problems, [68] the most common of which is that of counting the number of ways of writing a given number n as an ordered sum of 1s and 2s (called compositions); there are F n+1 ways to do this (equivalently, it's also the number of domino tilings of the rectangle).