Irrational Number Properties

Real Analysis · Axiom Academy

LESSON Irrational Number Properties A number no fraction can name: proving is irrational, and seeing why irrationals fill every gap on the line. 1. A Number No Fraction Can Name Plot on the number line. We can find fractions that get arbitrarily close — — yet not one of them ever lands exactly on . That stubborn gap is what makes the number irrational. Rational: a finite ratio of integers Irrational: no such ratio exists Suppose, for contradiction, that written in lowest terms — so , no common factor. Watch each consequence force the next, until the assumption destroys itself. Irrationals are not a thin scattering of oddities — they are dense in the reals. Between any two numbers , no matter how close, there is an irrational. Watch the interval shrink and an irrational keep appearing inside. Why the construction works: a rational plus a nonzero rational times an irrational is still irrational. Since , the point lands strictly between a and b , and it is irrational whenever a, b are rational. Always irrational. So stays irrational — the construction never accidentally lands on a fraction. Not necessarily irrational. For example . Closure fails — handle these case by case. You've seen what makes a number irrational, proved it for , and watched irrationals fill every gap on the real line. Where this leads: numbers like are algebraic (roots of integer polynomials), while and e are transcendental — irrational and not even algebraic.

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