Which list shows an accurate hierarchy of number sets within the real number system, from largest to smallest?
Responses
rational numbers, integers, whole numbers, natural numbers
rational numbers, integers, whole numbers, natural numbers - not selected, this is the correct answer
integers, rational numbers, natural numbers, whole numbers
integers, rational numbers, natural numbers, whole numbers - no response given
natural numbers, whole numbers, integers, rational numbers
natural numbers, whole numbers, integers, rational numbers - incorrect
irrational numbers, square roots, endless decimals, pi
irrational numbers, square roots, endless decimals, pi - incorrect
Which list shows an accurate hierarchy of number sets within the real number system, from largest to smallest?
rational numbers, integers, whole numbers, natural numbers
integers, rational numbers, natural numbers, whole numbers
natural numbers, whole numbers, integers, rational numbers
irrational numbers, square roots, endless decimals, pi
Please explain shortly
The correct hierarchy of number sets within the real number system, from largest to smallest, is:
rational numbers, integers, whole numbers, natural numbers.
Rational numbers include all numbers that can be expressed as a fraction or ratio of two integers.
Integers include both positive and negative whole numbers, including zero.
Whole numbers include all positive integers and zero.
Natural numbers include all positive integers, excluding zero.
The other options listed do not follow this correct hierarchy.
can you make it shorter
Sure! The correct hierarchy of number sets is:
Rational numbers > Integers > Whole numbers > Natural numbers.
Please explain shortly
To which sets of real numbers does zero belong?
Responses
whole numbers, integers, and irrational numbers
natural numbers, integers, and rational numbers
natural numbers, integers, and irrational numbers
whole numbers, integers, and rational numbers
Zero belongs to the sets of real numbers: whole numbers, integers, and rational numbers.
Explanation:
- Zero is included in the set of whole numbers as it is a non-negative integer, including zero itself.
- Zero is also included in the set of integers as it can be represented as 0/1, which is a ratio of two integers.
- Zero is part of the set of rational numbers as it can be expressed as a ratio of two integers (0/1).
Using a number line, what is the approximate value of 19−−√
4.25
4.13
4.5
4.38
Please explain shortly
The approximate value of √19 on a number line is 4.38.
Explanation:
To find the approximate value of √19 on a number line, we need to locate a value between which two integers the square root falls. When estimating, we know that 19 is closer to 16 (4²) than 25 (5²). So, we can estimate that √19 is slightly greater than 4. Therefore, the closest approximation is 4.38, which is closer to 4.38 than any of the other given options.
Please explain
Using rational approximations, what statement is true?
Responses
16 −−−√<4
12 −−−√<π
16−−√>4
12 −−−√ >π
Using rational approximations, the statement that is true is:
16 −−−√>4
Explanation:
When using rational approximations, it means representing a non-repeating decimal as a fraction. Since the square root of 16 is 4, which is already a rational number (4 can be written as 4/1), the statement "16 −−−√>4" is true. It is not necessary to use π or compare with irrational numbers in this case.
Write the expanded form of 357.25
Responses
(3×100)+(5×10)+(7×1)+(2×110)+(5×11,000)
open paren 3 times 100 close paren plus open paren 5 times 10 close paren plus open paren 7 times 1 close paren plus open paren 2 times 1 tenth close paren plus open paren 5 times the fraction with numerator 1 and denominator 1 comma 000 close paren - no response given
(3×100)+(5×10)+(7×1)+(2×1100)
open paren 3 times 100 close paren plus open paren 5 times 10 close paren plus open paren 7 times 1 close paren plus open paren 2 times 1 over 100 close paren - no response given
(3×100)+(5×10)+(7×1)+(2×110)+(5×1100)
open paren 3 times 100 close paren plus open paren 5 times 10 close paren plus open paren 7 times 1 close paren plus open paren 2 times 1 tenth close paren plus open paren 5 times 1 over 100 close paren - not selected, this is the correct answer
(3×100)+(5×10)+(7×1)+(5×110)+(2×1100)
The expanded form of 357.25 is:
(3 × 100) + (5 × 10) + (7 × 1) + (2 × 0.1) + (5 × 0.001)
Explanation:
To expand a decimal number, we break it down into its place values. In this case, we have:
- The digit 3 in the hundreds place, which is worth 3 × 100.
- The digit 5 in the tens place, which is worth 5 × 10.
- The digit 7 in the ones place, which is worth 7 × 1.
- The digit 2 in the tenths place, which is worth 2 × 0.1 (or 1/10).
- The digit 5 in the thousandths place, which is worth 5 × 0.001 (or 1/1000).
Combining these values gives us the expanded form of 357.25 as (3 × 100) + (5 × 10) + (7 × 1) + (2 × 0.1) + (5 × 0.001).
What is the expanded form of 8,471.716?
Responses
(8×1,000)+(4×100)+(7×10)+(1×1)+(7×110)+(1×1100)
open paren 8 times 1 comma 000 close paren plus open paren 4 times 100 close paren plus open paren 7 times 10 close paren plus open paren 1 times 1 close paren plus open paren 7 times 1 tenth close paren plus open paren 1 times 1 over 100 close paren - no response given
(8×103)+(4×102)+(7×101)+(1×100)+(7×1101)+(1×1102)+(6×1103)
(8×103)+(4×102)+(7×101)+(1×100)+(7×1101)+(1×1102)+(6×1103)
- no response given
(8×1,000)+(4×100)+(7×10)+(1×1)+(7×11)+(1×110)+(6×1100)
(8×1,000)+(4×100)+(7×10)+(1×1)+(7×11)+(1×110)+(6×1100)
- no response given
(8×1,000)+(4×100)+(7×10)+(1×1)+(7×110)+(1×1100)+(6×11,000
The expanded form of 8,471.716 is:
(8 × 1,000) + (4 × 100) + (7 × 10) + (1 × 1) + (7 × 0.1) + (1 × 0.01) + (6 × 0.001)
Explanation:
To expand the decimal number 8,471.716, we break it down into its place values. In this case, we have:
- The digit 8 in the thousands place, which is worth 8 × 1,000.
- The digit 4 in the hundreds place, which is worth 4 × 100.
- The digit 7 in the tens place, which is worth 7 × 10.
- The digit 1 in the ones place, which is worth 1 × 1.
- The digit 7 in the tenths place, which is worth 7 × 0.1 (or 1/10).
- The digit 1 in the hundredths place, which is worth 1 × 0.01 (or 1/100).
- The digit 6 in the thousandths place, which is worth 6 × 0.001 (or 1/1000).
Combining these values gives us the expanded form of 8,471.716.
To convert 0.264¯¯¯¯¯¯¯¯
to a rational number, you would set x
equal to 0.264¯¯¯¯¯¯¯¯
and then multiply both sides of the equation by what number?
Responses
999
999 - incorrect
1,000
1,000 - not selected, this is the correct answer
100
100 - no response given
10
explain
To convert the repeating decimal 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by 1000.
Explanation:
Since the repeating decimal 0.264¯¯¯¯¯¯¯¯ has the pattern of 264 repeating infinitely, we can represent it as the fraction x = 0.264¯¯¯¯¯¯¯¯. To eliminate the repeating part, we need to multiply both sides of the equation by a suitable number. In this case, multiplying by 1000 will shift the decimal point three places to the right, allowing us to get rid of the repeating part.
By multiplying both sides by 1000, we get 1000x = 264.264264... (without the repeating part).
Simplifying the equation, we have:
1000x = 264.264264...
x = 0.264264...
The resulting value of x, which is 0.264264..., is a rational number since it can be expressed as a fraction (e.g., 264/1000 = 33/125). Thus, by multiplying both sides of the equation by 1000, we can convert 0.264¯¯¯¯¯¯¯¯ to a rational number.
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator?
Responses
999
999 - not selected, this is the correct answer
1,000
1,000 - no response given
99
99 - incorrect
100
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
Explanation:
When a repeating decimal has a repeating cycle of three digits, we can set up an equation to convert it into a fraction. Let's say the repeating decimal is represented as x = 0.xyzxyzxyz..., where xyz is the repeating cycle of three digits.
To eliminate the repeating part, we multiply both sides of the equation by 1000 (since there are three digits in the repeating cycle):
1000x = xyz.xyzxyzxyz...
By subtracting the original equation from the above equation, we can eliminate the repeating part:
1000x - x = xyz.xyzxyzxyz... - 0.xyzxyzxyz...
999x = xyz
x = xyz/999
This means that the repeating decimal can be transformed into a fraction with a denominator of 999. Therefore, if a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
Convert 1650
to a decimal and a percentage.
Responses
34 and 3,400%
34 and 3,400% - no response given
0.32 and 32%
0.32 and 32% - not selected, this is the correct answer
3.125 and 312.5%
3.125 and 312.5% - no response given
0.32 and 3.2%
0.32 and 3.2% - incorrect
Score
To convert 1650 to a decimal and a percentage:
Decimal: 1650 is already in decimal form, so the decimal representation is 1650.
Percentage: To convert 1650 to a percentage, we divide it by 100 and multiply by 100 to get the percentage value.
(1650/100) * 100 = 165,000%
Therefore, the correct answer is:
Decimal: 1650
Percentage: 165,000%
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bot that is wrong
Using rational approximations, what statement is true?
Responses
16 −−−√<4
16 −−−√<4
- incorrect
12 −−−√<π
12 −−−√<π
- no response given
16−−√>4
square root of 16 is greater than 4 - no response given
12 −−−√ >π
12 −−−√ >π
- not selected, this is the correct answer
Using rational approximations, the true statement is:
12 −−−√ > π
Explanation:
When using rational approximations, we can compare the value of the square root with a rational number like π (pi). Since the square root of 12 is approximately 3.464, and π (pi) is approximately 3.14159, we can conclude that the square root of 12 is greater than π. Hence, the statement "12 −−−√ > π" is true.