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Name: Jehanzeb
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Date: 9/25/2005

Is the sum of two irrational numbers equal to a rational number?

The sum of two rational numbers, p/q + n/m = (pm +qn)/qm. If either p/q or n/m = X or Y, respectively, and either X or Y cannot be expressed as a ratio, then the sum cannot be expressed as a ratio. However, an irrational number can be expressed as an INFINITE sum of rational numbers. This was first noted by Newton (1676), but Euler gave the first proof (1774). So the sum of two IRRATIONAL numbers can be expressed as an INFINITE SUM of rational numbers.

Vince Calder


Two irrational numbers very RARELY add up to a rational number.

Any integer, finite decimal, or repeating decimal is a rational number. A rational number can be represented as a fraction. An irrational number cannot. It IS true that two RATIONAL numbers add up to a rational number: two fractions always add up to a fraction. Here are two examples: (1/2)+(1/3)=(5/6), 1+3=4. The latter applies because you can represent it as (1/1)+(3/1)=(4/1).

However, sqrt(2) is not rational because there is no fraction, no ratio of integers, that will equal sqrt(2). It calculates to be a decimal that never repeats and never ends. The same can be said for sqrt(3). Also, there is no way to write sqrt(2)+sqrt(3) as a fraction. In fact, the representation is already in its simplest form. To get two irrational numbers to add up to a rational number, you need to add irrational numbers such as [1+sqrt(2)] and [1-sqrt(2)]. In this case, the irrational portions just happen to cancel out, leaving: [1+sqrt(2)]+[1-sqrt(2)]=2. 2 is a rational number (i.e. 2/1).

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

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