Fractions

A fraction is any ratio between whole numbers. In mathematical terms, it is called a rational number. An irrational number is any number such as π which cannot be represented as the ratio of two whole numbers.

In today's blog, I will go over a single lemma regarding fractions:

Lemma: for any given rational number let's say a/b, there exists an integer let's say c such that: absolute(a/b - c) ≤ (1/2).

(1) To prove this, we need only consider the case where abs(a) is greater abs(b). [If abs(a) ≤ abs(b), then the conclusion follows from Corollary 2.1, here]

(2) From the division algorithm (see Theorem 1, here), we know that a = bq + r where r ≥ 0 and less than abs(b).

[For example, if a=-3, b=-2, then q=2, r=1 where r is greater than b but less than abs(b).]

(3) Let a'=abs(a), b'=abs(b)

(4) We know that a' - b'q is less than b' (since a' - b'q = r and r is less than b')

(5) So, it follows that: a'/b' - q is less than 1.

(6) Now, if both a,b are positive or a,b are negative, it follows that a/b = a'/b' and abs(a/b - q) is less than 1.

(7) If a,b are of different sign, than -a/b = a'/b' and -a/b - q = -(a/b + q) so that abs(a/b + q) is less than 1.

(8) The conclusion follows from Lemma 2, here.

QED

Corollary: if a/b is a rational number, then there exists an integer c such that: absolute(a/b - c/2) ≤ (1/4).

(1) From the lemma above, we know that for 2*(a/b), there exists a number c such that:
abs(2*(a/b) - c) ≤ (1/2).

(2) Dividing both sides by 2, gives us:
abs(a/b - c/2) ≤ (1/4)

QED

Now, it turns out that any number with a repeating decimal can be represented as a rational number.

Let me start with an example

(1) Let's assume that we have a decimal such as 5.234523452345... We can represent this decimal as a repeating decimal such as 5.2345.

(2) Now, we know if we multiply the number by 10⁴ we get:

52345.2345

(3) So, subtracting (2) by (1) gives us:

10⁴ - 1 = 52345 - 5 = 52340.

(4) So, the rational form of this repeating decimal is:

52340/9999.

Now, let's look at the proof that demonstrates this:

Lemma: Any number with a repeating decimal is rational.

(1) Any number with a repeating decimal can be represented with the following form:

d₁...d_m.a₁...a_n

NOTE: If a number has a nonrepeating portion, then we multiply this number by the number of nonrepeating digits, to get a number of the above form. Later, we divide our result by this same number.

(2) We can get an integer result by subtracting 10ⁿ*the number by the original number which after canceling for the repeating decimal gives us:

10ⁿ * (d₁...d_m.a₁...a_n...) - (d₁...d_m.a₁...a_n...) =
(d₁..d_ma₁..a_n) - (d₁..d_m).

(3) Now, our rational number is equal to the value in step #2 divided by 10ⁿ - 1.

QED