0
Does there exist an irrational number such that its square root is rational?
Wiki User
∙ 10y ago
No, and I can prove it:
-- The product of two rational numbers is always a rational number.
-- If the two numbers happen to be the same number, then it's
the square root of their product.
-- Remember ... the product of two rational numbers is always a rational number.
-- So the square of a rational number is always a rational number.
-- So the square root of an irrational number can't be a rational number
(because its square would be rational etc.).
Wiki User
∙ 10y ago
Add your answer:
Earn +20 pts
What is the difference between rational numbers and irrational numbers?
An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. A rational number can be.A rational number is defined to be a number that can be expressed as the ratio of two integers. An irrational number is any real number that is not rational. A rational number is a number that can be expressed as a fraction. An irrational number is one that can not.Some examples of rational numbers would be 5, 1.234, 5/3, or -3Some examples of irrational numbers would be π, the square root of 2, the golden ratio, or the square root of 3.A rational number is a number that either has a finite end or a repeating end, such as .35 or 1/9 (which is .1111111 repeating).An irrational number has an infinite set of numbers after the decimal that never repeat, such a the square root of 2 or pi.A rational number is one that can be expressed as a ratio of two integers, x and y (y not 0). An irrational number is one that cannot be expressed in such a form.In terms of decimal numbers, a rational number has a decimal representation that is terminating or [infinitely] recurring. The decimal representation for an irrational is neither terminating nor recurring. (Recurring decimals are also known as repeating decimals.)A rational number is a number that can be expressed as a fraction. An irrational number is one that can not.Some examples of rational numbers would be 5, 1.234, 5/3, or -3Some examples of irrational numbers would be π, the square root of 2, the golden ratio, or the square root of 3.An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. A rational number can be.A rational number can be represented by a ratio of whole numbers. An irrational number cannot. There are many more irrational numbers than there are rational numbersRational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.A rational number can be expressed as a fraction, with integers in the numerator and the denominator. An irrational number can't be expressed in that way. Examples of irrational numbers are most square roots, cubic roots, etc., the number pi, and the number e.A rational number can always be written as a fractionwith whole numbers on the top and bottom.An irrational number can't.A rational number can always be written as a fraction with whole numbers on top and bottom.An irrational number can't.Any number that you can completely write down, with digits and a decimal pointor a fraction bar if you need them, is a rational number.A rational number can be expressed as a fraction whereas an irrational can not be expressed as a fraction.Just look at the definition of a rational number. A rational number is one that can be expressed as a fraction, with integers (whole numbers) in the numerator and the denominator. Those numbers that can't be expressed that way - for example, the square root of 2 - are said to be irrational.A rational number is any number that can be written as a ratio or fraction. If the decimal representation is finite orhas a repeating set of decimals, the number is rational.Irrational numbers cannot be reached by any finite use of the operators +,-, / and *. These numbers include square roots of non-square numbers, e.g.√2.Irrational numbers have decimal representations that never end or repeat.Transcendental numbers are different again - they are irrational, but cannot be expressed even with square roots or other 'integer exponentiation'. They are the numbers in between the numbers between the numbers between the integers. Famous examples includee or pi (π).By definition: a rational number can be expressed as a ratio of two integers, the second of which is not zero. An irrational cannot be so expressed.One consequence is that a rational number can be expressed as a terminating or infinitely recurring decimal whereas an irrational cannot.This consequence is valid whatever INTEGER base you happen to select: decimal, binary, octal, hexadecimal or any other - although for non-decimal bases, you will have the "binary point" or "octal point" in place of the decimal point and so on.A rational number can be expressed as a fraction whereas an irrational number can't be expressed as a fractionRational numbers can be expressed as a ratio of two integers, x/y, where y is not 0. Conventionally, y is taken to be greater than 0 but that is not an essential element of the definition. An irrational number is one for which such a pair of integers does not exist.Rational numbers can be expressed as one integer over another integer (a "ratio" of the two integers) whereas irrational numbers cannot.Also, the decimal representation ofa rational number will either: terminate (eg 31/250 = 0.124); orgo on forever repeating a sequence of digits at the end (eg 41/330 = 0.1242424... [the 24 repeats]);whereas an irrational number will not terminate, nor will there be a repeating sequence of digits at the end (eg π = 3.14159265.... [no sequence repeats]).Rational numbers are numbers that keeps on going non-stop, for example pie. Irrational numbers end. Its as simple as that! Improved Answer:-Rational numbers can be expressed as fractions whereas irrational numbers can't be expressed as fractions.a rational number can be expressed as a fraction in the form a/b (ie as a fraction).a irrational number cannot be expressed as a fraction (e.g. pi, square root of 2 etc)Rational numbers can be represented as fractions.That is to say, if we can write the number as a/b where a and b are any two integers and b is not zero. If we cannot do this, then the number is irrational.For example, .5 is a rational number because we can write it as 5/10=1/2The square root of 2 is irrational because there do not exist integers a and b suchthat square root of 2 equals a/b.Rational numbers can be expressed as fractions whereas irrational numbers can't be expressed as fractions.
What is the square root of -70?
Ordinarily, the square root of -70 does not exist since a positive number squared or a negative number squared results in a positive number. If complex numbers are permitted, then the square root of -70 is i√70 ~= 8.367i where i is the imaginary square root of -1 such that i2 = -1.
Prove that if a and b are rational numbers then a plus b is a rational number?
Because a is rational, there exist integers m and n such that a=m/n. Because b is rational, there exist integers p and q such that b=p/q. Consider a+b. a+b=(m/n)+(p/q)=(mq/nq)+(pn/mq)=(mq+pn)/(nq). (mq+pn) is an integer because the product of two integers is an integer, and the sum of two integers is an integer. nq is an integer since the product of two integers is an integer. Because a+b equals the quotient of two integers, a+b is rational.
Why properties of square roots are true only for non negative numbers?
The square of a "normal" number is not negative. Consequently, within real numbers, the square root of a negative number cannot exist. However, they do exist within complex numbers (which include real numbers)and, if you do study the theory of complex numbers you wil find that all the familiar properties are true.
Square root of -1 is iota How?
Mathematicians decided that, since the square root of a negative number does not exist, they would use the first letter of "imaginary" to represent this "value".
Is the square root of an irrational number rational?
No: Let r be some irrational number; as such it cannot be represented as s/t where s and t are both non-zero integers. Assume the square root of this irrational number r was rational. Then it can be represented in the form of p/q where p and q are both non-zero integers, ie √r = p/q As p is an integer, p² = p×p is also an integer, let y = p² And as q is an integer, q² = q×q is also an integer, let x = q² The number is the square of its square root, thus: (√r)² = (p/q)² = p²/q² = y/x but (√r)² = r, thus r = y/x and is a rational number. But r was chosen to be an irrational number, which is a contradiction (r cannot be both rational and irrational at the same time, so it cannot exist). Thus the square root of an irrational number cannot be rational. However, the square root of a rational number can be irrational, eg for the rational number ½ its square root (√½) is not rational.
What are examples of irrational numbers?
An example is the square root of a number. Ex: square root of 2. This is 1 example, not the main one. Any cube root or square root which doesn't give a perfect number is an irrational number. Ex; square root and cube root of 5, since their answer will be 2.24 and 1.70 which are not perfect numbers like square roots of 25 and 64 or cube roots of 27 and 216.
What is the difference between rational numbers and irrational numbers?
An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. A rational number can be.A rational number is defined to be a number that can be expressed as the ratio of two integers. An irrational number is any real number that is not rational. A rational number is a number that can be expressed as a fraction. An irrational number is one that can not.Some examples of rational numbers would be 5, 1.234, 5/3, or -3Some examples of irrational numbers would be π, the square root of 2, the golden ratio, or the square root of 3.A rational number is a number that either has a finite end or a repeating end, such as .35 or 1/9 (which is .1111111 repeating).An irrational number has an infinite set of numbers after the decimal that never repeat, such a the square root of 2 or pi.A rational number is one that can be expressed as a ratio of two integers, x and y (y not 0). An irrational number is one that cannot be expressed in such a form.In terms of decimal numbers, a rational number has a decimal representation that is terminating or [infinitely] recurring. The decimal representation for an irrational is neither terminating nor recurring. (Recurring decimals are also known as repeating decimals.)A rational number is a number that can be expressed as a fraction. An irrational number is one that can not.Some examples of rational numbers would be 5, 1.234, 5/3, or -3Some examples of irrational numbers would be π, the square root of 2, the golden ratio, or the square root of 3.An irrational number is a number that can't be expressed by a fraction having integers in both its numerator and denominator. A rational number can be.A rational number can be represented by a ratio of whole numbers. An irrational number cannot. There are many more irrational numbers than there are rational numbersRational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.A rational number can be expressed as a fraction, with integers in the numerator and the denominator. An irrational number can't be expressed in that way. Examples of irrational numbers are most square roots, cubic roots, etc., the number pi, and the number e.A rational number can always be written as a fractionwith whole numbers on the top and bottom.An irrational number can't.A rational number can always be written as a fraction with whole numbers on top and bottom.An irrational number can't.Any number that you can completely write down, with digits and a decimal pointor a fraction bar if you need them, is a rational number.A rational number can be expressed as a fraction whereas an irrational can not be expressed as a fraction.Just look at the definition of a rational number. A rational number is one that can be expressed as a fraction, with integers (whole numbers) in the numerator and the denominator. Those numbers that can't be expressed that way - for example, the square root of 2 - are said to be irrational.A rational number is any number that can be written as a ratio or fraction. If the decimal representation is finite orhas a repeating set of decimals, the number is rational.Irrational numbers cannot be reached by any finite use of the operators +,-, / and *. These numbers include square roots of non-square numbers, e.g.√2.Irrational numbers have decimal representations that never end or repeat.Transcendental numbers are different again - they are irrational, but cannot be expressed even with square roots or other 'integer exponentiation'. They are the numbers in between the numbers between the numbers between the integers. Famous examples includee or pi (π).By definition: a rational number can be expressed as a ratio of two integers, the second of which is not zero. An irrational cannot be so expressed.One consequence is that a rational number can be expressed as a terminating or infinitely recurring decimal whereas an irrational cannot.This consequence is valid whatever INTEGER base you happen to select: decimal, binary, octal, hexadecimal or any other - although for non-decimal bases, you will have the "binary point" or "octal point" in place of the decimal point and so on.A rational number can be expressed as a fraction whereas an irrational number can't be expressed as a fractionRational numbers can be expressed as a ratio of two integers, x/y, where y is not 0. Conventionally, y is taken to be greater than 0 but that is not an essential element of the definition. An irrational number is one for which such a pair of integers does not exist.Rational numbers can be expressed as one integer over another integer (a "ratio" of the two integers) whereas irrational numbers cannot.Also, the decimal representation ofa rational number will either: terminate (eg 31/250 = 0.124); orgo on forever repeating a sequence of digits at the end (eg 41/330 = 0.1242424... [the 24 repeats]);whereas an irrational number will not terminate, nor will there be a repeating sequence of digits at the end (eg π = 3.14159265.... [no sequence repeats]).Rational numbers are numbers that keeps on going non-stop, for example pie. Irrational numbers end. Its as simple as that! Improved Answer:-Rational numbers can be expressed as fractions whereas irrational numbers can't be expressed as fractions.a rational number can be expressed as a fraction in the form a/b (ie as a fraction).a irrational number cannot be expressed as a fraction (e.g. pi, square root of 2 etc)Rational numbers can be represented as fractions.That is to say, if we can write the number as a/b where a and b are any two integers and b is not zero. If we cannot do this, then the number is irrational.For example, .5 is a rational number because we can write it as 5/10=1/2The square root of 2 is irrational because there do not exist integers a and b suchthat square root of 2 equals a/b.Rational numbers can be expressed as fractions whereas irrational numbers can't be expressed as fractions.
How are rational numbers different than irrational numbers?
Rational NumbersA rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. The number 8 is a rational number because it can be written as the fraction 8/1.Likewise, 3/4 is a rational number because it can be written as a fraction.Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1.Irrational NumbersAll numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:π = 3.141592… = 1.414213…Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!
An integer that is not rational?
The are an infinite number of rational numbers that are not integers, because a rational number is a number that is written as a ratio of two integers. For examble, 1/2 (i.e. a half) is a non-integer rational number. This form is generally called a fraction.
Does there exist a quadratic equation whose coefficients are irrational but both the roots are rational?
None, if the coefficients of the quadratic are in their lowest form.
Why is there no direct proof of the infinity of primes naturals sqrt2 after 2000 years?
A direct proof of the infinity of primes would require what is essentially a formula to calculate the Nth prime number; such a formula isn't even guaranteed to exist. It's possible to formulate a proof of the infinity of primes that would be, in a sense, direct. A direct proof that the square root of 2 is irrational is impossible, because the irrational numbers aren't defined in any direct way - just as the real numbers which aren't rational. So to prove that the square root of 2 is irrational, we have to prove that it's not rational, which requires indirect techniques.
Why does advertising exist and what products seem to work best with rational and irrational advertising campaigns?
Advertising exists to help businesses improve
What is the longest rational number?
There are infinitely many rational numbers and, in decimal form, most of them have infinitely many digits. So there cannot be a longest rational number.
Is 4.761 an irrational number?
No. It is a finite decimal number.An irrational number is one that cannot be expressed as a ratio between two integers, such as pi (which has infinite digits) or square roots of numbers that are not squares of other numbers.An imaginary number is a different type, such as sqrt (-1), which cannot exist in actuality but appears as a value in higher mathematics.
Where would you use a negative rational number?
Anywhere where negative quantities exist.
What is smallest rational number?
There is no such number. If any number laid claim to being the smallest rational number its claim could be challenged by half that number - which would also be rational and, obviously smaller. And the claim of that number could be challenged by half that number, and so on.
