Which statement is true about the relationships between the number sets?(1 point) Responses All rational numbers are also integers. All rational numbers are also integers. Whole numbers include all natural numbers and 0. Whole numbers include all natural numbers and 0. Some irrational numbers are also integers. Some irrational numbers are also integers. Not all natural numbers are real numbers.
Not all natural numbers are real numbers.
The statement "Not all natural numbers are real numbers" is true.
To determine which statement is true about the relationships between number sets, we will review each statement and explain the concepts involved.
1. "All rational numbers are also integers."
Explanation: Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. Integers are whole numbers that can be both positive and negative, including zero. This statement implies that every rational number is also an integer. However, this statement is false because rational numbers can be non-integer values, such as 1/2 or -3/4.
2. "Whole numbers include all natural numbers and 0."
Explanation: Whole numbers are similar to integers but do not include negative numbers. They start from zero and include all positive integers. Natural numbers are counting numbers, starting from 1 onward. This statement is true because whole numbers include all natural numbers and zero.
3. "Some irrational numbers are also integers."
Explanation: Irrational numbers are numbers that cannot be expressed as fractions or terminated decimals. They are non-repeating and non-terminating decimal numbers, such as π (pi) or √2 (square root of 2). Integers, as stated earlier, are whole numbers, including both positive and negative numbers. This statement is false because irrational numbers cannot be integers since they are non-integer values.
4. "Not all natural numbers are real numbers."
Explanation: Natural numbers, as mentioned before, are counting numbers starting from 1 onward. Real numbers encompass all numbers, including both rational and irrational numbers. This statement is false because all natural numbers are included in the set of real numbers.
Based on the explanations provided, the correct statement is: "Whole numbers include all natural numbers and 0."
Which of the following is true about −9 ?(1 point) Responses It is a whole number but not an integer. It is a whole number but not an integer. It is both an integer and a whole number. It is both an integer and a whole number. It is an integer but not a whole number. It is an integer but not a whole number. It is an integer but not a rational number. It is an integer but not a rational number.
The statement "It is both an integer and a whole number" is true about -9.
A sign in a gas station advertises gas at the rate of $3.39910 per gallon. How many dollars does a gallon cost, writing the price strictly as a decimal?(1 point)
To write the price strictly as a decimal, we can round the given price to the desired number of decimal places.
In this case, rounding to 2 decimal places, the price per gallon would be $3.40.
Which set of numbers is always rational?(1 point) Responses fractions fractions integers integers positive numbers positive numbers decimals
The set of numbers that is always rational is 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?(1 point)
To find the approximate height of an equilateral triangle, we can substitute the given values into the formula.
Given:
Side length = 8 meters
Approximation for the square root of 3 (sqrt(3)) = 74
The formula for the height of an equilateral triangle is:
Height = sqrt(3) * (Side length)
Substituting the given values:
Height = 74 * 8
Height = 592 meters
Therefore, the approximate height of an equilateral triangle with a side length of 8 meters is 592 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)
To find the approximate length from one corner to an opposite corner of a square park, we can use the given values in the formula.
Given:
Side length of the square park = 700 feet
Approximation for the square root of 2 (sqrt(2)) = 1.4
The formula for the diagonal of a square is:
Diagonal = sqrt(2) * (Side length)
Substituting the given values:
Diagonal = 1.4 * 700
Diagonal = 980 feet
Therefore, the approximate length from one corner to an opposite corner of a square park that is 700 feet on a side is 980 feet.
Using a number line, what whole number and rational number to the nearest half is 40−−√ between?(1 point) 40−−√ is between and .
To determine the whole number and rational number to the nearest half that 40-√ is between, we need to find the two nearest numbers on the number line.
Given:
40-√
Approximation for the square root of 40 (√40) = 6.32
Nearest whole number to 6.32 is 6.
Nearest whole number to 7.0 is 7.
Nearest half below 6.32 is 6.
Nearest half above 6.32 is 6.5.
Therefore, 40-√ is between 6 and 6.5 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 8 and 8.5 between 7.5 and 8 between 7.5 and 8 between 8.5 and 9 between 8.5 and 9 between 7 and 7.5
To determine the whole number and rational number to the nearest half that 58-√ is between, we need to find the two nearest numbers on the number line.
Given:
58-√
Approximation for the square root of 58 (√58) = 7.62
Nearest whole number to 7.62 is 8.
Nearest half below 7.62 is 7.5.
Nearest half above 7.62 is 8.
Therefore, 58-√ is between 7.5 and 8 on the number line.
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 convert both radicals to rational numbers.
Given:
Approximation for the square root of 81 (√81) = 9
Approximation for the square root of 8 (√8) = 2.83
The inequality statement comparing the two expressions is:
9 > 2.83
Therefore, the inequality 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 can divide 811 by the desired denominator. Since there is no specific denominator mentioned, I will assume it to be 1.
Dividing 811 by 1:
811 ÷ 1 = 811
Since the dividend (811) is already an integer and there is no remainder, the decimal representation of 811 is not a repeating decimal. It remains the same as the whole number.
Therefore, 811 expressed as a decimal is simply 811.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert the repeating decimal 0.72¯¯¯¯¯ to a rational number in simplest form, we can use the concept of fractions.
Let's represent the repeating decimal 0.72¯¯¯¯¯ as x.
To remove the repeating part, we can multiply both sides of the equation by a power of 10 that is equal to the number of decimal places in the repeating part. In this case, there is one digit repeating, so we will multiply by 10.
10x = 7.2¯¯¯¯¯
By subtracting the original equation, we can eliminate the repeating part:
10x - x = 7.2¯¯¯¯¯ - 0.72¯¯¯¯¯
9x = 6.48
To obtain the simplest form, we divide both sides of the equation by 9:
9x/9 = 6.48/9
x = 0.72
Therefore, the repeating decimal 0.72¯¯¯¯¯ can be expressed as a rational number in simplest form as 0.72.
Which of the following is a rational number between 5 and 514 ?(1 point) Responses 517 5 Start Fraction 1 over 7 end fraction 513 5 Start Fraction 1 over 3 end fraction 26−−√ start root 26 end root 525 5 Start Fraction 2 over 5 end fraction
Among the options provided, the rational number between 5 and 514 is 26−√.
Therefore, the answer is: 26−√.
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 need to convert the radicals into rational numbers.
Given:
Approximation for the square root of 7 (√7) = 2.65
Approximation for the square root of 8 (√8) = 2.83
Now, we can find the rational number with a denominator of 7 between these two approximations.
Between 7-√ and 8-√, the rational number with a denominator of 7 that falls in between is 1/7.
Therefore, the answer is: 1/7, expressed as an improper fraction.
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.(4 points)
The relationship between rational numbers and whole numbers can be described as follows:
1. All rational numbers are not necessarily whole numbers:
Rational numbers are numbers that can be expressed as fractions, where the numerator and denominator are both integers. Whole numbers, on the other hand, include all natural numbers (positive integers) and zero. Not all rational numbers can be whole numbers because they can include non-integer values. For example, 1/2 (a rational number) is not a whole number.
2. All whole numbers are rational numbers:
Whole numbers can be expressed as fractions with a denominator of 1. For example, 3 as a fraction would be 3/1. Since the numerator and denominator in this case are both integers, all whole numbers can also be considered rational numbers.
In summary, while all whole numbers are rational numbers, not all rational numbers are whole numbers. Whole numbers are a subset of rational numbers, and rational numbers include both whole numbers and non-integer values.
Simplify that
The relationship between rational numbers and whole numbers can be simplified as follows:
1. All rational numbers are not whole numbers.
2. All whole numbers are rational numbers.
In simpler terms, while all whole numbers are rational numbers, not all rational numbers are whole numbers.