#MathsInMinutes Day 15: Irrational Numbers
● Irrational numbers are numbers that cannot be expressed by dividing one natural number by another.
● Unlike rational numbers, they cannot be expressed as a ratio between two integers, or in a decimal form that either comes to an end or lapses into a regular pattern of repeating digits.
● Instead, the decimal expansions of irrational numbers carry on forever without periodic repetition.
● Like the natural numbers and the rationals, the irrationals are infinite in extent.
● But whilst the rationals and the integers are sets of the same size, or cardinality, the irrationals are far more numerous still.
● In fact their nature makes them not only infinite, but uncountable.
● Some of the most important numbers in mathematics are irrational, including π, the ratio between the circumference or a circle and its radius, Euler´s constant e, the golden ratio shown in the next slide, and √2, the square root of 2.
● The golden ratio is the ratio between two numbers when the ratio of the smaller one to the larger is equal to the ratio of the larger to the sum of the whole. It is an irrational number and a constant that arises naturally in many situations and is used to govern proportion in art and architecture.
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