Which of the following is irrational?
V3/4
53.781654292
3.6363636363636363…
-7 8/4
Explain why.
The irrational numbers are numbers that cannot be expressed as a fraction or a ratio of two integers.
Out of the given options, the number 3.6363636363636363... is irrational.
To explain why, we can understand that this decimal representation continues indefinitely with the pattern of repeating decimal digits. Any repeating decimal can be converted into a fraction, so if 3.6363636363636363... were rational, we could express it as a fraction. However, since the decimal continues indefinitely with the repeating pattern, there is no finite fraction that represents this number. Thus, 3.6363636363636363... is irrational.
Wrong.
Apologies for the incorrect response.
The correct answer is:
Out of the given options, the number 3.6363636363636363... is irrational.
To explain why, we can understand that this decimal representation continues indefinitely with the pattern of repeating decimal digits. Any repeating decimal can be converted into a fraction, so if 3.6363636363636363... were rational, we could express it as a fraction. However, we can simplify the repeating decimal as follows:
Let x = 3.636363...
Then 100x = 363.636363...
By subtracting the equation "100x = 363.636363..." from "x = 3.636363...", we get:
99x = 360
x = 360/99
Since 360 and 99 share no common factors other than 1, the fraction 360/99 is in its simplest form and cannot be further reduced. Therefore, 3.636363... is a repeating decimal that cannot be expressed as a fraction of two integers, making it irrational.
The answer is v3/4 explain why
Apologies for the confusion. Let's discuss why V3/4 is irrational.
To determine if V3/4 is rational or irrational, we need to check if we can simplify it into a fraction of two integers.
To simplify V3/4, we need to find a fraction where the numerator and denominator have no common factors other than 1.
Let's assume that V3/4 is a rational number and can be written as a fraction p/q, where p and q are integers and have no common factors other than 1.
Then, by squaring both sides, we get:
(V3/4)^2 = (p/q)^2
3/16 = p^2/q^2
Cross-multiplying, we have:
3q^2 = 16p^2
From this equation, we can observe that since 3 is a prime number, it should also be a factor on the left side of the equation. However, this is not the case for 16p^2. This implies that 3 cannot be a factor of 16p^2, which contradicts our assumption that p/q is a fraction with no common factors other than 1.
Therefore, V3/4 cannot be expressed as a fraction of two integers, indicating that it is an irrational number.
To determine which of the following numbers is irrational, we need to understand the definition of an irrational number. An irrational number cannot be expressed as a fraction (or ratio) of two integers and does not terminate or repeat indefinitely.
Let's examine the given options:
1. V3/4:
This number is rational because it can be expressed as a fraction. Rational numbers can be written as the ratio of two integers, and V3/4 can be simplified to (√3)/2.
2. 53.781654292:
This number is rational since it can be expressed as a decimal, which terminates after a finite number of digits.
3. 3.6363636363636363…:
This number is rational since it can be expressed as a repeating decimal. It can be written as 3 and then a repeating block of "36."
4. -7 8/4:
This number is rational because it can be expressed as a mixed number, which is a combination of an integer and a fraction.
Therefore, the only option that is irrational is not provided among the given choices.
To determine which of the given numbers is irrational, we need to understand the concept of irrational numbers. An irrational number is a real number that cannot be expressed as a fraction, where the decimal representation goes on forever without repeating or terminating.
Let's analyze each of the given numbers:
1. V3/4: This is the square root of 3 divided by 4. To determine if it is rational or irrational, we need to simplify the fraction. Since the square root of 3 is not a perfect square, it cannot be expressed as a fraction. Therefore, V3/4 is irrational.
2. 53.781654292: This number is a terminating decimal, which means it has a finite number of digits after the decimal point. Terminating decimals can always be expressed as fractions, making this number rational.
3. 3.6363636363636363…: This number has a repeating decimal pattern where the digits 36 repeat. Repeating decimals can also be expressed as fractions. In this case, the repeating pattern can be represented as the fraction 36/99, which simplifies to 4/11. Therefore, 3.6363636363636363… is rational.
4. -7 8/4: This is a mixed number, which is a combination of a whole number and a fraction. Mixed numbers can always be expressed as fractions. In this case, -7 8/4 can be rewritten as -7 + 8/4 = -7 + 2 = -5. Since -5 can be expressed as a fraction (-5/1), it is rational.
To summarize, the irrational numbers among the given options are V3/4.