The last chapter, 23, contains details about some number curiosities like lucky mistakes, including anomalous cancellations, Pascal’s triangle, and Pythagorean triples. Lucky mistakes are mistakes that result in correct answers despite adopting the wrong operation. An anomalous cancellation is the illogical cancellation of digits or groups of digits in the numerator and denominator to reduce a fraction, producing a correct result. For example, when cancelling as many ‘84’ as desired, the fractions, \( \frac{484}{847} \) , \( \frac{48484}{84847} \) , \( \frac{4848484}{8484847} \) , …, \( \frac{\kern0.62em 4\dots \dots ..\mathrm{84}}{\kern0.62em 84\dots .\dots .7} \) , always give the correct result, that is, 4/7. Infinite families of such lucky fractions, including the lucky sum and product of fractions, are discussed in this chapter. Apart from patterns and properties of Pascal’s triangle, Fibonacci numbers, perfect numbers, π, and e from Pascal’s triangle are covered. Some Pythagorean triplet curiosities including Fermat’s problem and Pythagorean triples using Fibonacci numbers are also dealt with in this chapter.

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Number Curiosities

  • Shyam Sunder Gupta

摘要

The last chapter, 23, contains details about some number curiosities like lucky mistakes, including anomalous cancellations, Pascal’s triangle, and Pythagorean triples. Lucky mistakes are mistakes that result in correct answers despite adopting the wrong operation. An anomalous cancellation is the illogical cancellation of digits or groups of digits in the numerator and denominator to reduce a fraction, producing a correct result. For example, when cancelling as many ‘84’ as desired, the fractions, \( \frac{484}{847} \) , \( \frac{48484}{84847} \) , \( \frac{4848484}{8484847} \) , …, \( \frac{\kern0.62em 4\dots \dots ..\mathrm{84}}{\kern0.62em 84\dots .\dots .7} \) , always give the correct result, that is, 4/7. Infinite families of such lucky fractions, including the lucky sum and product of fractions, are discussed in this chapter. Apart from patterns and properties of Pascal’s triangle, Fibonacci numbers, perfect numbers, π, and e from Pascal’s triangle are covered. Some Pythagorean triplet curiosities including Fermat’s problem and Pythagorean triples using Fibonacci numbers are also dealt with in this chapter.