Any number that can be found in the real world is, literally, a real number. Counting objects gives a sequence of positive integers, or natural numbers, If you consider having nothing or being in debt as a number, then the set of integers, including zero and negative numbers, is in order. If you cut a cake into equal pieces, then you may have a piece which represents a rational number, which is a number that can be represented by an irreducible fraction of two integers. What is the next? We have the set of real numbers, which is the union of the set of rational numbers and the set of irrational numbers. The Venn diagram below depicts the relationship between these sets of numbers.
is the set of numbers that can be measured, such as length or weight, which, of course, include which can be obtained from counting, subtracting, and dividing. What could be an example of number which is not in Presumably, the first irrational number in history is which was found by a follower of Pythagoras. corresponds to the length of the diagonal of a square with side length 1, so is a real number, which exists in the real world. However, cannot be represented by an irreducible fraction of two integers. We can prove it here shortly.
Prove that is not a rational number.
Let where and are coprime integers. Then the square of is which implies that is a multiple of 2. If the square of a number is a multiple of 2, then the number is also a multiple of 2. Therefore, using the substitution where is an integer, gives Then and is also a multiple of 2. Now we have both and as multiples of 2, and this does not correspond with the condition that and are coprime integers. Hence we can conclude that cannot be represented by an irreducible fraction of two integers, so it is an irrational number.
Likewise, measuring objects may give numbers other than rational numbers. Other common examples are the ratio of a circle's circumference to its diameter and Euler's number. Meanwhile, the exact definition or construction of is quite difficult to understand, and it is taught in college-level math courses. For your information, is commonly constructed by Dedekind cuts.
Clarify which set is a subset of another set among and --the sets of irrational numbers, natural numbers, rational numbers, real numbers, and integers, respectively.
The set of natural numbers is the set of positive integers.
The set of integers is the set of rational numbers with denominator 1.
The set of rational numbers is the set of real numbers which can be represented by irreducible fractions of two integers.
The set of irrational numbers is the set of real numbers which are not rational.
Is it true that
No, is not an element of but is an irrational number. We can prove it similarly with the case of
Let where and are coprime integers. Then the square of is which implies is a multiple of 3. If the square of a number is a multiple of 3, then the number is also a multiple of 3. Therefore, substituting where is an integer, gives Then and is also a multiple of 3. Now we have both and as multiples of 3, which does not correspond with the condition that and are coprime integers. Hence we can conclude that cannot be represented by an irreducible fraction of two integers, so it is an irrational number.
Is it true that
Yes, is an element of
By definition, is a positive number that satisfies We know that so which is an integer. All integers are rational numbers, so we can conclude that
The length of the longer side of an A4 paper is 29.7 cm. Is 29.7 rational?
We can represent 29.7 as an irreducible fraction of two integers:
Therefore, 29.7 is rational.
The area of an equilateral triangle with side length 1 is Is rational?
First, let's prove that the product of a nonzero rational number where and are coprime integers, and an irrational number cannot be rational. Assume that where and are coprime integers. Then
where and Now we have as a fraction of two integers, which contradicts with the premise that is irrational.
Therefore since is irrational, a fourth of is also irrational.
The circumference of a circle with radius 1 is Is rational?
As we have proved above, the product of a nonzero rational number and an irrational number cannot be rational. is known to be irrational. Hence twice is also irrational.
Which of the following statements are wrong?
- (a) Each point on the real line corresponds to a unique real number.
- (b) There are more real numbers between any two consecutive integers than the whole set of integers.
- (c) We can't say (b) above since we can't compare two infinities.
- (d) Every bounded interval is a finite set.
- (e) Some bounded intervals are finite sets.
- (f) There are finite irrational numbers between any two given irrational numbers.