Which list shows an accurate hierarchy of number sets within the real number system, from largest to smallest? (1 point)
A:integers, rational numbers,
natural numbers, whole numbers
B:irrational numbers, square roots, endless decimals, pi
C:natural numbers, whole
numbers, integers, rational
numbers
D: rational numbers, integers,
whole numbers, natural
numbers
D: rational numbers, integers, whole numbers, natural numbers
Describe the set of numbers that are real numbers but cannot be expressed as one integer divided by another.(1 point)
rational
rational
irrational
irrational
negative
negative
improper
The set of numbers that are real numbers but cannot be expressed as one integer divided by another are irrational numbers. These numbers cannot be expressed as a fraction or ratio of two integers. Examples of irrational numbers include pi (∏), square roots of non-perfect squares (such as √2 or √5), and infinite non-repeating decimals (such as 0.101001000100001...).
Describe the set of numbers that are real numbers but cannot be expressed as one integer divided by another.(1 point)
Responses
A:rational
B:irrational
C:negative
D:improper
B: irrational
To which sets of real numbers does zero belong?(1 point)
A:natural numbers, integers, and rational numbers
B:whole numbers, integers, and irrational numbers
C:natural numbers, integers, and irrational numbers
D:whole numbers, integers, and rational numbers
A: natural numbers, integers, and rational numbers
A cube has an edge length, a, that is a rational number of centimeters. Identify which measurement will require the use of irrational numbers.
A:the diagonal of one of the cube’s faces
B:the volume of the cube
C:the area of one of the cube’s faces
D:the surface area of the cube
A: the diagonal of one of the cube's faces.
To determine the accurate hierarchy of number sets within the real number system, we need to understand the definitions of each set.
1. Natural numbers (N): These are the counting numbers, starting from 1 and going infinitely: 1, 2, 3, 4, ...
2. Whole numbers (W): The set of whole numbers includes all natural numbers, as well as zero (0): 0, 1, 2, 3, 4, ...
3. Integers (Z): The set of integers includes all whole numbers, as well as their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
4. Rational numbers (Q): Rational numbers can be expressed as fractions, where the numerator and denominator are both integers. They include terminating decimals (such as 0.5) and recurring decimals (such as 1.333...).
5. Irrational numbers (I): These numbers cannot be expressed as fractions and their decimal representations neither terminate nor repeat. Examples include √2, π (pi).
Now, let's analyze each option provided:
A: Integers, rational numbers, natural numbers, whole numbers
This option is not accurate because it places integers (Z) before rational numbers (Q), which is incorrect.
B: Irrational numbers, square roots, endless decimals, pi
This option is also inaccurate because it does not include the other number sets like natural numbers, whole numbers, integers, and rational numbers.
C: Natural numbers, whole numbers, integers, rational numbers
This option correctly arranges the number sets in the hierarchy from largest to smallest. It starts with natural numbers (N) and goes all the way to rational numbers (Q).
D: Rational numbers, integers, whole numbers, natural numbers
This option is incorrect because it reverses the order between natural numbers (N) and rational numbers (Q). Natural numbers come before rational numbers in the hierarchy.
Therefore, the correct answer is option C: natural numbers, whole numbers, integers, rational numbers.