In your own words, explain why the sum of a rational number and an irrational number is irrational. Provide an example with your explanation.
To understand why the sum of a rational number and an irrational number is always irrational, let's first define what a rational and an irrational number are.
A rational number is any number that can be expressed as a fraction, where the numerator and denominator are integers, and the denominator is not zero. For example, 1/2, -3/4, and 7/1 are all rational numbers.
On the other hand, an irrational number is any real number that cannot be expressed as a fraction or a ratio of integers. These numbers have non-repeating and non-terminating decimal representations. Some common examples include √2 (square root of 2), π (pi), and e.
Now, let's consider the sum of a rational number and an irrational number. We'll use the example of 2/3 (rational) + √2 (irrational).
We can start by assuming that the sum, let's call it x, is rational. This means we can write x as a fraction p/q, where p and q are integers. So, we have:
x = p/q
Now, let's rearrange the equation to isolate the irrational term:
√2 = x - 2/3
Now, we square both sides to eliminate the square root:
2 = (x - 2/3)^2
Expanding this equation, we get:
2 = x^2 - (4/3)x + 4/9
Rearranging and simplifying, we have:
3x^2 - (4/3)x + (2 - 4/9) = 0
Multiplying through by 9 to eliminate fractions, we get:
27x^2 - 12x + 18 - 4 = 0
27x^2 - 12x + 14 = 0
Now, let's analyze the quadratic equation we obtained. If it has rational roots (solutions), then x would be rational. However, using the discriminant (b^2 - 4ac) of the quadratic equation, we can determine whether the roots are rational or irrational.
The discriminant is given by:
D = (-12)^2 - 4(27)(14) = 144 - 1512 = -1368
Since the discriminant is negative, the quadratic equation has no rational roots, which means the sum of the rational number and the irrational number is irrational. In this example, the sum of 2/3 and √2 is irrational.
So, in general, when adding a rational number to an irrational number, the result is always irrational due to the nature of irrational numbers and the algebraic manipulation required to analyze their sum.