Rational and Irrational Numbers Practice
Math 8 Q1 (Pre-Algebra) / The Number System
Which of the following is a rational number: 2π, 60/61−−√, or 0.37815?(1 point)'
The rational number is 60/61-√.
Which of the following is a rational number: 2π, 6/0 ,61−−√, or 0.37815?(1 point)'
The rational number in the given options is 0.37815.
Rational and Irrational Numbers Practice
Math 8 Q1 (Pre-Algebra) / The Number System
The circumference of a circle equals pi times the diameter. Using 3.14 to approximate pi, what is the approximate circumference in inches of a circle with a diameter of 1/2 in.? (1 point)
To find the circumference of a circle, you multiply pi times the diameter.
Given that the diameter is 1/2 in, we need to find the circumference using the value of pi as 3.14.
Circumference = pi * diameter
Circumference = 3.14 * (1/2)
Simplifying this expression, we get:
Circumference = 3.14/2
Calculating the value, we obtain:
Circumference ≈ 1.57 inches
94 is close to the square root of what integer?
To determine which of the given numbers is rational, we need to understand the definition of rational and irrational numbers.
Rational numbers are numbers that can be expressed as the quotient, or fraction, of two integers. In other words, they can be written in the form p/q, where p and q are integers and q is not equal to 0.
On the other hand, irrational numbers cannot be expressed as a fraction of two integers. They are non-repeating and non-terminating decimals. Examples of irrational numbers include √2, π, and e.
Now let's analyze the given options:
1. 2π: This is a product of 2 and π. Since π is an irrational number, multiplying it by any integer will still result in an irrational number. Therefore, 2π is also an irrational number.
2. 60/61−−√: This expression involves a square root of a fraction, which can be a bit tricky. To simplify it, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of 61−−√ is 61+−−√.
Applying the conjugate to the fraction, we get:
(60/61−−√) * (61+−−√)/(61+−−√)
Simplifying the numerator, we have:
(60 * (61+−−√))/(61+−−√)
Expanding the numerator, we get:
(60 * 61) + (60 * −−√)
= 3660 + 60−−√
So, 60/61−−√ can be simplified to the expression 3660 + 60−−√. This is still an irrational number.
3. 0.37815: This is a decimal number. To determine if it is rational or irrational, we need to determine if it is a non-repeating and non-terminating decimal. One way to do this is to convert the decimal to a fraction. If the decimal can be expressed as a fraction, it is rational.
Converting 0.37815 to a fraction:
Let x = 0.37815.
Then 10000x = 3781.5. Subtracting x from both sides, we get:
10000x - x = 3781.5 - 0.37815
Simplifying both sides:
9999x = 3781.12185
Dividing both sides by 9999, we have:
x = 3781.12185/9999
Although the fraction 3781.12185/9999 is not in simplified form, it shows that 0.37815 can be expressed as a fraction. Therefore, 0.37815 is a rational number.
To summarize, out of the given options, the number 0.37815 is the only one that is rational.