Describe the set of numbers that are real numbers but cannot be expressed as one integer divided by another.(1 point)
The set of real numbers that cannot be expressed as one integer divided by another is known as the irrational numbers. These are numbers that cannot be written as a fraction or ratio of two integers. Some famous examples of irrational numbers include $\sqrt{2}$, $\pi$, and $e$.
To which sets of real numbers does zero belong?(1 point)
Zero belongs to the following sets of real numbers:
1. Whole Numbers: Zero is included in the set of whole numbers, which consists of all positive integers (starting from zero), their negatives, and zero itself ($\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$).
2. Integers: Zero is a member of the set of integers, which includes all whole numbers and their negatives ($\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$).
3. Rational Numbers: Zero can be expressed as the ratio of two integers, specifically $0 = \frac{0}{1}$, so it is also a rational number.
4. Real Numbers: Zero is a real number, as it falls on the number line between the negative and positive 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.
The measurement that will require the use of irrational numbers is the diagonal of the cube.
The diagonal of a cube is a line segment that connects two opposite vertices. Let's label the vertices of the cube as A, B, C, D, E, F, G, and H. If we consider the edge length of the cube to be a, then the diagonal can be represented by the line segment connecting points A and E (or B and F, C and G, D and H).
Using the Pythagorean theorem, we can find the length of the diagonal, labeled as d:
$d^2 = a^2 + a^2 + a^2$
$d^2 = 3a^2$
$d = \sqrt{3a^2}$
Since the edge length, a, is assumed to be a rational number, the square root of 3 will introduce an irrational number into the measurement of the diagonal. Hence, the measurement of the diagonal will require the use of irrational numbers.
simplify that
Apologies for the confusion in the previous response. Let's simplify the expression further.
Using the Pythagorean theorem, we have:
$d^2 = 3a^2$
To simplify, we can take the square root of both sides:
$\sqrt{d^2} = \sqrt{3a^2}$
$d = \sqrt{3}a$
The simplified expression for the length of the diagonal, d, of the cube in terms of the edge length, a, is $d = \sqrt{3}a$.
Which of the following numbers could be used to specify how many cups of flour should be used in a bread recipe?(1 point)
The following numbers could be used to specify how many cups of flour should be used in a bread recipe:
- 1
- 1/2
- 3/4
- 2
These numbers are all rational numbers and can easily represent a specific quantity of cups of flour.