Question
Prove that 
is an irrational number.
The correct answer is: Yes, it's an irrational number
Hint:
The real numbers which cannot be expressed in the form of p/q(where p and q are integers and q ≠ 0) are known as irrational numbers. In the given question we are asked to prove if is irrational number. To do so we use the method of contradiction.
Solution
Let's assume that is a rational number.
Step 1 of 2:
If is rational, that means it can be written in the form of p/q, where p and q integers that have no common factor other than 1 i.e. they are coprime than 1 and q ≠ 0.
So,
q = p
Squaring both sides,
5q2 = p2
q2=
This means p2 can be divided by 5
Step 2 of 2:
So if p2 can be divided by 5, then p can also be divided by 5
So, let’s say
p=5r
Squaring both sides
Putting the value of p2 in eq-1
So, this means q² is divisible by 5 and so q is also divisible by 5.
Final Answer: p and q both are divisible 5 but we assumed that p and q are coprime and this contradiction has arisen because of our incorrect assumption that is a rational number. So, we conclude that is an irrational number.
q = pThis means p2 can be divided by 5
Step 2 of 2:
So if p2 can be divided by 5, then p can also be divided by 5
Final Answer: p and q both are divisible 5 but we assumed that p and q are coprime and this contradiction has arisen because of our incorrect assumption that
is a rational number. So, we conclude that is an irrational number.Related Questions to study
State if 0.24758326…. is a rational number or irrational number with a suitable explanation.
Note:
Irrational numbers are those real numbers which cannot be expressed in the form of p/q(where p and q are integers and q ≠ 0). So in this question, if we try to express the given decimal expansion in p/q form, we will not be able to do that. Hence, we can directly say that the given number is an irrational number.
State if 0.24758326…. is a rational number or irrational number with a suitable explanation.
Note:
Irrational numbers are those real numbers which cannot be expressed in the form of p/q(where p and q are integers and q ≠ 0). So in this question, if we try to express the given decimal expansion in p/q form, we will not be able to do that. Hence, we can directly say that the given number is an irrational number.
Prove that is an irrational number.
Prove that is an irrational number.
The line in the xy-plane above represents the relationship between the height h(x), in feet, and the base diameter x , in feet, for cylindrical Doric columns in ancient Greek architecture. How much greater is the height of a Doric column that has a base diameter of 5 feet than the height of a Doric column that has a base diameter of 2 feet?
The line in the xy-plane above represents the relationship between the height h(x), in feet, and the base diameter x , in feet, for cylindrical Doric columns in ancient Greek architecture. How much greater is the height of a Doric column that has a base diameter of 5 feet than the height of a Doric column that has a base diameter of 2 feet?
The table above shows shipping charges for an online retailer that sells sporting goods. There is a linear relationship between the shipping charge and the weight of the merchandise. Which function can be used to determine the total shipping charge f(X), in dollars, for an order with a merchandise weight of x pounds?
The table above shows shipping charges for an online retailer that sells sporting goods. There is a linear relationship between the shipping charge and the weight of the merchandise. Which function can be used to determine the total shipping charge f(X), in dollars, for an order with a merchandise weight of x pounds?
A television with a price of 
is to be purchased with an initial payment 
of and weekly payments of 
. Which of the following equations can be used to find the number of weekly payments,
, required to complete the purchase, assuming there are no taxes or fees?
A television with a price of is to be purchased with an initial payment of and weekly payments of . Which of the following equations can be used to find the number of weekly payments, , required to complete the purchase, assuming there are no taxes or fees?
2Z+1=Z
What value of Z satisfies the equation above?
2Z+1=Z
What value of Z satisfies the equation above?
The image which represents the table is,
| Color |
Children |
| Red |
8 |
| Blue |
7 |
| Green |
5 |
| Yellow |
9 |
| Pink |
6 |
The image which represents the table is,
| Color |
Children |
| Red |
8 |
| Blue |
7 |
| Green |
5 |
| Yellow |
9 |
| Pink |
6 |
The product of -2.655 × 18.44 is _________.
1)Write the given decimal as a fraction by adding the number one to the denominator. The given value of the numerator and denominator is to be multiplied by multiples of 10 to remove the decimal(.) value. If 1.8, for example, is a decimal value, 18/10 will be the corresponding fraction. We cannot simply carry on with 18/10. 2)The specified decimal should first be written as a ratio (p/q), with one as the denominator.
3)The numerator and denominator are to be multiplied by multiples of 10 to convert the decimal into a whole number, multiplied for each decimal point. (Multiply by 100/100 if two numbers follow the decimal point)
4) just the outcome fraction
The product of -2.655 × 18.44 is _________.
1)Write the given decimal as a fraction by adding the number one to the denominator. The given value of the numerator and denominator is to be multiplied by multiples of 10 to remove the decimal(.) value. If 1.8, for example, is a decimal value, 18/10 will be the corresponding fraction. We cannot simply carry on with 18/10. 2)The specified decimal should first be written as a ratio (p/q), with one as the denominator.
3)The numerator and denominator are to be multiplied by multiples of 10 to convert the decimal into a whole number, multiplied for each decimal point. (Multiply by 100/100 if two numbers follow the decimal point)
4) just the outcome fraction
A square is inscribed in a circle with radius 
inches. What is the perimeter of the square in inches?
A square is inscribed in a circle with radius inches. What is the perimeter of the square in inches?
What value of x satisfies the equation above?
What value of x satisfies the equation above?
What positive value of x satisfies the equation above?
What positive value of x satisfies the equation above?
If (x, y) is the solution to the given system of equations, what is the value of y ?
If (x, y) is the solution to the given system of equations, what is the value of y ?
The table gives some values of x and the corresponding values of f(x) for polynomial function f. Which of the following could be the graph of f in the xy - plane, where y = f(x)?
| x | f(x) |
| -2 | -2 |
| -1 | 3 |
| 0 | 2 |
| 1 | 7 |
| 2 | 30 |
The table gives some values of x and the corresponding values of f(x) for polynomial function f. Which of the following could be the graph of f in the xy - plane, where y = f(x)?
| x | f(x) |
| -2 | -2 |
| -1 | 3 |
| 0 | 2 |
| 1 | 7 |
| 2 | 30 |
, where m and b are constants, is shown in the xy - plane. What is the value of m ?
, in grams, of the substance 
years after 200 grams of the substance is applied to the aquatic environment? (The half-life is the length of time needed for an amount of the substance to decrease to one-half of that amount.)
