Which statement is true about the relationships between the number sets?(1 point)
Responses
Some irrational numbers are also integers.
All rational numbers are also integers.
Not all natural numbers are real numbers.
Whole numbers include all natural numbers and 0.
Whole numbers include all natural numbers and 0.
Which of the following is true about −9?(1 point)
Responses
It is an integer but not a rational number.
It is a whole number but not an integer.
It is an integer but not a whole number.
It is both an integer and a whole number.
Which set of numbers is always rational?
The set of numbers that is always rational is the set of integers. Integers can be expressed as fractions, where the denominator is always 1.
Which set of numbers is always rational?(1 point)
Responses
positive numbers
decimals
fractions
integers
The set of numbers that is always rational is the set of fractions.
The height of an equilateral triangle equals 3√2 times the side length. Using 74 for the square root of 3, what is the approximate height in meters of an equilateral triangle with a side length of 8 meters?
Given that the height of an equilateral triangle equals 3√2 times the side length, we can substitute 74 for the square root of 3.
Height = 3√2 * side length
Height = 3 * 74 * 2 * 8
Calculating this expression:
Height = 3 * 74 * 2 * 8
Height = 1776 meters
Therefore, the approximate height of the equilateral triangle with a side length of 8 meters is 1776 meters.
The diagonal of a square equals 2–√ times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side?(1 point)
Given that the diagonal of a square equals 2–√ times the side length, we can substitute 1.4 for the square root of 2.
Diagonal = (2–√) * side length
Diagonal = (2–1.4) * 700
Calculating this expression:
Diagonal = (2–1.4) * 700
Diagonal = 0.6 * 700
Diagonal = 420 feet
Therefore, the approximate length from one corner to an opposite corner of the square park is 420 feet.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point)
To determine the whole number and rational number to the nearest half that 40−√ lies between, we can approximate the value of 40−√ and determine its nearest whole number and nearest half on a number line.
The square root of 40 is approximately 6.324, so 40−√ is approximately 40 - 6.324 = 33.676.
Nearest Whole Number: The nearest whole number to 33.676 is 34.
Nearest Half: To determine the nearest half, we can look at the distance between 33.676 and the nearest whole numbers, which are 33 and 34.
The distance between 33.676 and 33 is 33.676 - 33 = 0.676
The distance between 33.676 and 34 is 34 - 33.676 = 0.324
Since 0.324 is smaller than 0.676, the nearest half is 33.5.
Therefore, 40−−√ is between the whole number 34 and the rational number 33.5 on the number line.
Using a number line, what whole number and rational number to the nearest half is 58−−√ between?
To determine the whole number and rational number to the nearest half that 58−√ lies between, we can approximate the value of 58−√ and determine its nearest whole number and nearest half on a number line.
The square root of 58 is approximately 7.616, so 58−√ is approximately 58 - 7.616 = 50.384.
Nearest Whole Number: The nearest whole number to 50.384 is 50.
Nearest Half: To determine the nearest half, we can look at the distance between 50.384 and the nearest whole numbers, which are 50 and 51.
The distance between 50.384 and 50 is 50.384 - 50 = 0.384
The distance between 50.384 and 51 is 51 - 50.384 = 0.616
Since 0.384 is smaller than 0.616, the nearest half is 50.5.
Therefore, 58−√ is between the whole number 50 and the rational number 50.5 on the number line.
Using a number line, what whole number and rational number to the nearest half is 58−−√ between?(1 point)
Responses
between 8 and 8.5
between 7.5 and 8
between 8.5 and 9
between 7 and 7.5
To determine the whole number and rational number to the nearest half that 58−√ lies between, we can approximate the value of 58−√ and determine its nearest whole number and nearest half on a number line.
The square root of 58 is approximately 7.6158, so 58−√ is approximately 58 - 7.6158 = 50.3842.
Nearest Whole Number: The nearest whole number to 50.3842 is 50.
Nearest Half: To determine the nearest half, we can look at the distance between 50.3842 and the nearest whole numbers, which are 50 and 51.
The distance between 50.3842 and 50 is 50.3842 - 50 = 0.3842.
The distance between 50.3842 and 51 is 51 - 50.3842 = 0.61583.
Since 0.3842 is smaller than 0.61583, the nearest half is 50.5.
Therefore, 58−√ is between the whole number 50 and the rational number 50.5 on the number line, so the correct response is "between 50 and 50.5".
Using rational approximations, what is an inequality statement comparing 81−−√ and 8–√8? Enter <, >, or =.(1 point)
The inequality is: 81−−√
8–√8.
To compare 81−√ and 8-√8 using rational approximations, we can approximate the values and compare them.
Approximation of 81−√:
√81 = 9
Therefore, 81−√ is approximately 81-9 = 72.
Approximation of 8-√8:
√8 is approximately 2.828
Therefore, 8-√8 is approximately 8 - 2.828 = 5.172
Comparing the approximations:
72 > 5.172
Therefore, the inequality statement comparing 81−√ and 8-√8 is: 81−√ > 8-√8
Express 811 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
To express 811 as a repeating decimal, we divide 811 by the denominator 9 (since the cycle length is 1 digit).
811 ÷ 9 = 90 remainder 1.
Next, we bring down a zero and divide 10 (the remainder multiplied by 10) by 9 again.
10 ÷ 9 = 1 remainder 1.
We continue this process, bringing down zeros and dividing the remainder by 9 until the remainder becomes 0 or until the repeating pattern becomes apparent.
The resulting expression is:
811 ÷ 9 = 90 and a repeating decimal pattern of 1, represented as follows:
811/9 = 90.1¯¯¯¯¯¯¯¯¯
The bar over the digit "1" indicates that it repeats infinitely.
onvert 0.72¯¯¯¯¯ to a rational number in simplest form
To convert 0.72 recurring to a rational number in simplest form, we can let x equal 0.72 recurring and use algebraic manipulation to find the rational form.
Step 1: Multiply both sides of the equation x = 0.727272... by 100 to remove the recurring part:
100x = 72.727272...
Step 2: Subtract the equation 100x = 72.727272... - x = 0.727272... from the previous equation:
100x - x = 72.727272... - 0.727272...
99x = 72
Step 3: Solve for x by dividing both sides of the equation by 99:
x = 72/99
Step 4: Simplify 72/99 by finding the greatest common divisor (GCD) of 72 and 99 and dividing both numerator and denominator by the GCD.
GCD(72, 99) = 9
72/9 = 8
99/9 = 11
Therefore, 0.72 recurring can be written as the rational number 8/11 in simplest form.
Which of the following is a rational number between 5 and 514?(1 point)
Responses
5 1/3
5 2/5
5 1/7
26−−√
To determine a rational number between 5 and 514, we can analyze the given options:
5 1/3: This mixed number is not between 5 and 514.
5 2/5: This mixed number is not between 5 and 514.
5 1/7: This mixed number is not between 5 and 514.
26−√: It is not given as a decimal or fraction, so we cannot compare it directly to the values of 5 and 514.
Therefore, none of the given options are rational numbers between 5 and 514.
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
To find a rational number with a denominator of 7 that is between 7−√ and 8−√, we can approximate their decimal values.
Approximation of 7−√:
√7 is approximately 2.646
Therefore, 7−√ is approximately 7 - 2.646 = 4.354
Approximation of 8−√:
√8 is approximately 2.828
Therefore, 8−√ is approximately 8 - 2.828 = 5.172
To find a rational number with a denominator of 7 between 4.354 and 5.172, we can select a decimal value that is between these two values and convert it to an improper fraction with a denominator of 7.
For example, 4.5 is between 4.354 and 5.172.
To convert 4.5 to an improper fraction with a denominator of 7:
4.5 = 9/2
Multiplying the numerator and denominator by 7 to change the denominator to 7:
(9/2) * (7/7) = 63/14
Therefore, a rational number with a denominator of 7 between 7−√ and 8−√ is 63/14.
Describe the relationship between rational numbers and whole numbers. Are all rational numbers also whole numbers? Are all whole numbers also rational numbers? Give examples to support your answer.
Rational numbers and whole numbers are related, but they are not the same.
All whole numbers are rational numbers. A whole number can be expressed as a fraction where the denominator is 1. For example, 5 can be written as 5/1, which is a fraction and thus a rational number.
However, not all rational numbers are whole numbers. Rational numbers include any number that can be expressed as a fraction of two integers, where the denominator is not zero. This means that rational numbers can include fractions and decimals. For example, 3/2 and 1.5 are both rational numbers, but they are not whole numbers.
So, to summarize:
- All whole numbers are rational numbers.
- Not all rational numbers are whole numbers.