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Which of the following is an Irrational Number

Which of the following is an Irrational Number

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Which of the following is an Irrational Number

Question 1

a) 4/55
b) 2.31312345….
c) 1.404040…..
d) 1.11

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The correct answer for the given question is Option b) 2.31312345….

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Question 2

A) √79
B) 3.25
C) √10000
D) None of these

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The correct answer for the given question is Option A) √79

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Irrational Numbers

Real numbers that cannot be represented by a ratio are called irrational numbers. In other words, irrational numbers are those numbers that are not rational. During the 5th century BC, Hippasus, a Pythagorean philosopher, discovered irrational numbers. However, he was ridiculed and thrown into the sea because of his theory. In mathematics, irrational numbers are real numbers which cannot be expressed as fractions, p/q, where p and q are integers. The denominator q is not equal to zero (q ≠ 0). Furthermore, the decimal expansion of an irrational number is neither terminating nor repeating.

Properties of Irrational Numbers

The characteristics of irrational numbers allow us to select them from a set of real numbers. The characteristics of irrational numbers are as follows:

  • Irrational numbers are composed of non-terminating and non-recurring decimal points.
  • These are real numbers only.
  • The result of adding an irrational number and a rational number is an irrational number only. A rational number y and an irrational number x result in x+y = an irrational number.
  • If two irrational numbers are multiplied by a nonzero rational number, their product is an irrational number. In the case of an irrational number x and a rational number y, their product xy = irrational.
  • Any two irrational numbers may or may not have a least common multiple (LCM).
  • Two irrational numbers might be rational when they are added, subtracted, multiplied, and divided.

History of Irrational Numbers

The Greek philosopher Hippassus of Metapontum is generally credited with being the first person to acknowledge irrational numbers. He is said to have tried to apply his teacher’s famous theorem to his problem a^{2}+b^{2}= c^{2} to find the length of the diagonal of a unit square. The length of the sides of a square cannot be expressed as the ratio of two integers. As the other Pythagoreans held that only positive rational numbers could exist, their subsequent actions have been the subject of speculation for centuries. The discovery may have led to Hippassus’ death.

Some Assumptions and Imagination

  • Pythagoreans believed that irrational numbers were so horrifying that Hippassus was thrown overboard on a voyage and vowed to keep the secret.
  • As a result of Hippassus’ discovery of irrational numbers, the Pythagoreans in Flanders ostracized him and the gods scuttled his boat on the high seas because they were disgusted.
  • The mathematician Hippassus discovered irrational numbers and then died in a natural accident while travelling by ocean (the sea is a treacherous place). The fact remains that his coworkers were still so dissatisfied with his discovery that they wished they had been the ones to throw him overboard.

The stories above may not be true, and they are stories that have been invented and embellished through time to illustrate a pivotal moment in history.

It is unclear, however, which method Hippassus used to discover irrational numbers. If you’re curious, the Brilliant summary of rational numbers leads up to Euclid’s irrational proof \sqrt{2}. . Hippassus may have achieved this in this way. Some scholars, however, believe that Euclid’s method (written 300 years after Hippassus) is more advanced than what Hippassus might have been able to accomplish.The act of proving an irrational number is difficult to imagine as a moral transgression, regardless of what actually happened.

Since the beginning of recorded history, humans have had numbers. It is the practical necessity to count and measure things that provides the earliest basis for numbers and math. The process of counting naturally produces positive, non-zero, natural numbers. In addition, one can easily imagine how measurement could be applied to things that couldn’t be divided into whole units or whose dimensions were between whole numbers. As ratios of the natural numbers, fractions were a practical invention. It was probably intuitive to discover the positive rational numbers. In addition to serving practical purposes, numbers also served as a spiritual foundation for Pythagorean philosophy and religion. “All is number” was the premise of Pythagorean cosmology, physics, ethics, and spirituality. They believed that everything, from stars in the sky to musical scales to virtues, can be understood and described by rational numbers.

Fyodor Bronnikov depicts a Pythagorean celebrating sunrise in a 19th-century painting. Pythagoreans believed that everything, from the sunrise to the musical harmony, contains rational numbers, which gave everything mystical significance.Since there is an infinite supply of positive rational numbers, this suggests they are the basis for everything in the universe. On the surface, it might seem that an infinite number of numbers would be enough to describe anything that exists.Rational numbers are incredibly dense on the number line. There is not much “space” between 1/100000 and 1/100001, but if you needed to describe something between those two numbers, you would have no trouble finding a fraction between them. The number line is infinitely dense with rational numbers. In addition to this infinite density, irrational numbers suggest that there are still gaps in the number line that are not represented by ratios of two integers.

By the time of Pythagorean civilization, diagonals of unit squares were probably manually measured. The measurement was probably considered an approximation that was close to the actual length of the diagonal.In the absence of Hippassus, they had no reason to suspect that there are logically real numbers that in principle, not just in practice, cannot be measured and counted. It would have been like discovering a void in the universe, if you had believed all numbers were rational numbers and that rational numbers were the basis of everything in the universe. It was a sign of meaninglessness in what had seemed to be an orderly world to see an irrational number. For the Pythagoreans, numbers ought to be something you can count on, and all things should be counted rationally. The discovery of an irrational number proved that things existed in the universe that were not comprehendible using rational numbers, which threatened not only Pythagorean mathematics but also their philosophy.

FAQs,

  • How many irrational numbers are there between 1 and 6 ?

There are infinite irrational numbers between 1 and 6. Irrational numbers have an infinite number of decimal places and cannot be expressed as fractions.It is possible to find an infinite number of irrational numbers between any two numbers on an array, regardless of the value or distance between them.

  • Is the sum of two irrational numbers always irrational?

The sum of two irrational numbers is sometimes irrational.The sum of two irrational numbers, in some cases, will be irrational. However, if the irrational parts of the numbers have a zero sum (cancel each other out), the sum will be rational.

  • Integers are _____ irrational numbers

A) Always
B) Sometimes
C) Never

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The correct answer for the given question is Option C) Never

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Which value is equal to 5% of 1,500?

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