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This is also a repeating binary fraction 0.0 0011... . It may come as a surprise that terminating decimal fractions can have repeating expansions in binary. It is for this reason that many are surprised to discover that 1/10 + ... + 1/10 (addition of 10 numbers) differs from 1 in binary floating point arithmetic. In fact, the only binary ...
Fractions are written as two integers, the numerator and the denominator, with a dividing bar between them. The fraction m / n represents m parts of a whole divided into n equal parts. Two different fractions may correspond to the same rational number; for example 1 / 2 and 2 / 4 are equal, that is:
The continued fraction representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for example 137 / 1600 = 0.085625, or infinite with a repeating cycle, for example 4 / 27 = 0.148148148148...
Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to , and 25.12 is equal to . [20] Every rational number corresponds to a finite or a repeating decimal. [21] [c]
Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
The number π (/ p aɪ /; spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159.The number π appears in many formulae across mathematics and physics.
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1]