Rational and irrational numbers definition
2 answers:
Rational numbers can be defined as numbers that can be written in fractional notation. Let's consider an arbitrary number 
, where a and b do not have common factors. This is an example of a rational number, as it can be described using a fraction. Real numbers and fractional numbers are among this branch of numbers.
Irrational numbers, on the other hand, are simply those numbers which cannot be written as an exact fraction, only approximated fractions. Famous cases of irrational numbers are those of π (pi), e (Euler's number/ Napier's constant), and many square root numbers.
Irrational numbers, on the other hand, are simply those numbers which cannot be written as an exact fraction, only approximated fractions. Famous cases of irrational numbers are those of π (pi), e (Euler's number/ Napier's constant), and many square root numbers.
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A 3d cardboard box has 6 sides, each of which are rectangles. If you unfold the 3D box, and flatten it out, then you'll be left with 6 rectangles such as what you see in the attachment below. This is one way to unfold the box. This flattened drawing is the net of the 3D rectangular prism. You can think of it as wrapping paper that covers the exterior of the box. There are no gaps or overlapping portions. If you can find the area of each piece of the net, and add up those pieces, that gets you the total area of the net. This is the exactly the surface area of the box.
In the drawing below, I've marked the sides as: top, bottom, left, right, front, back. This way you can see how the 3D box unfolds and how the sides correspond to one another. Other net configurations are possible.
Let's call the two numbers and
.
Given these variables, we can say: , based on the first sentence in the problem.
Also, remember that the reciprocal of a number is simply 1 divided by the number. Thus, we can say that:
To solve, we can simply substitute in for
in the second equation and solve.
- Set up
- Get terms on the left side to a common denominator for easier addition
- Simplify
- Cross multiplication (
)
- Simplify
- Subtract
from both sides of the equation
- Factor left side of the equation
- Zero Product Property
Now, notice that we have found two solutions, but the problem is only asking for one. This <em>likely </em>means that one of our solutions is extraneous. Let's take a look. Remember that the smaller positive number is equal to 14 less than the larger number. However,
,
Since is not positive in this case,
is not a solution.
Thus, is our only solution. In this case,
,
which means that the smaller number is 14 and the larger number is 28.