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 Math Help > Sets, Set theory, Number systems > Sets > Rational and Irrational

In every closed interval with distinct endpoints, there is at least one rational number and one irrational number.

# A Rational and an Irrational in Every Closed Interval

(1) Every closed interval that contains more than one point (that is, its endpoints are distinct) contains a rational number and an irrational number.

(2) Between every pair of rational numbers there is an irrational number.  And

(3) Between every pair of irrational numbers there is a rational number.

The arguments for these facts depend, in turn, on some more facts:

(4) The sum of rational + rational is rational

proof: a/b + c/d = (ad+bc)/(bd)

(5) The sum of rational + irrational is irrational

proof: if a is rational and b is irrational and a+b=c is rational then c+(-a)=b is rational, a contradiction

(6) The product of rational * rational is rational

proof: (a/b)(c/d) = (ab)/(cd)

(7) The product of rational * irrational is irrational as long as the rational is not zero

proof: if a is non-zero rational and b is irrational and ab=c is rational then c(1/a) is rational, a contradiction

Now, let's find an irrational number b that lies between any two rational numbers a and c, where a < c:

Let b = a + (c-a)*sqrt(2)/2 is
rational + rational*irrational, so it is irrational.

a < a + (c-a)*sqrt(2)/2 < a + (c-a) = c, so
a < b < c, and b is irrational, so statement (2) is proven

Now, let's find a rational number b that lies between any two irrational numbers a and c, with a < c:

If c-a < 1, then let n = ceiling(-log10(c-a))
If c-a ≥ 1, then let n = 0

Now let a' = a*10n+1, and let c' = c*10n+1

c' - a' ≥ 10, so
a' < ceiling(a') < floor(c') < c', so
a = a'/10n+1 < ceiling(a')/10n+1 < floor(c')/10n+1 < c'/10n+1 = c

Let b = floor(c')/10n+1, so b is a rational number, and

a  < b < c, so statement (3) is proven.

Now, let's tackle statement (1): that every closed interval with distinct endpoints contains a rational number and an irrational number.

Let the endpoints be a and c.

If a is rational and c is irrational, or vice-versa, then statement (1) is true.  Now let's assume they're either both rational or both irrational.

If a and c are both rational, then statement (2) shows there is also an irrational number between them, proving (1).

If a and c are both irrational, then statement (3) shows there is also a rational number between them, proving (1).

In fact, a statement stronger than (1) can be proven:

(8) Every interval, open or closed, that contains more than one point (that is, its endpoints are distinct) contains an infinite number of rationals and an infinite number of irrationals.

Proof: given two real numbers, a and f such that a < f, then let b=(4a+f)/5, c=(3a+2f)/5, d=(2a+3f)/5, and e=(a+4f)/5

Then a < b < c < d < e < f, so the interval between a and f contains two disjoint intervals [b,c] and [d,e]

Since every interval contains two disjoint closed intervals, every interval contains an infinite number of disjoint closed intervals.

Since each closed interval contains a rational number and an irrational number, every interval contains an infinite number of rational numbers and an infinite number of irrational numbers, proving statement (8).

### Related pages in this website

Definition of Interval

Subdividing an Interval

Calculus Theorems

The webmaster and author of this Math Help site is Graeme McRae.