Math and Arithmetic

Numbers

Irrational Numbers

Top Answer

It will be irrational. Rational numbers are numbers that can be written as a fraction. Irrational Numbers cannot be expressed as a fraction.

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0The sum of the three can be rational or irrational.

The sum of a rational and irrational number must be an irrational number.

No. In fact the sum of a rational and an irrational MUST be irrational.

Such a sum is always irrational.

It is always irrational.

The value of the sum depends on the values of the rational number and the irrational number.

The sum is irrational.

It is an irrational number.

Any, and every, irrational number will do.

Since the sum of two rational numbers is rational, the answer will be the same as for the sum of an irrational and a single rational number. It is always irrational.

The sum of two irrational numbers may be rational, or irrational.

It is always an irrational number.

It is always an irrational number.

Wrong. It is always an irrational number.

No - the sum of any two rational numbers is still rational:

Let R1 = rational number Let X = irrational number Assume R1 + X = (some rational number) We add -R1 to both sides, and we get: -R1 + x = (some irrational number) + (-R1), thus X = (SIR) + (-R1), which implies that X, an irrational number, is the sum of two rational numbers, which is a contradiction. Thus, the sum of a rational number and an irrational number is always irrational. (Proof by contradiction)

Let `a` be a rational number and `b` be an irrational number,assume that the sum is rational. 1.a +b =c Where a and c are rational and b is irrational. 2.b=c-a Subtracting the same number a from each side. 3.b is irrational c-a is a rational number we arrived at a contradiction. So the sum is an irrational number.

No. The sum of an irrational number and any other [real] number is irrational.

It may be a rational or an irrational number.

The sum of a rational and an irrational number is always irrational. Here is a brief proof:Let a be a rational number and b be an irrational number, and c = a + b their sum. By way of contradiction, suppose c is also rational. Then we can write b = c - a. But since c and a are both rational, so is their difference, and this means that bis rational as well. But we already said that b is an irrational number. This is a contradiction, and hence the original assumption was false. Namely, the sum c must be an irrational number.

Yes Yes, the sum of two irrational numbers can be rational. A simple example is adding sqrt{2} and -sqrt{2}, both of which are irrational and sum to give the rational number 0. In fact, any rational number can be written as the sum of two irrational numbers in an infinite number of ways. Another example would be the sum of the following irrational quantities [2 + sqrt(2)] and [2 - sqrt(2)]. Both quantities are positive and irrational and yield a rational sum. (Four in this case.) The statement that there are an infinite number of ways of writing any rational number as the sum of two irrational numbers is true. The reason is as follows: If two numbers sum to a rational number then either both numbers are rational or both numbers are irrational. (The proof of this by contradiction is trivial.) Thus, given a rational number, r, then for ANY irrational number, i, the irrational pair (i, r-i) sum to r. So, the statement can actually be strengthened to say that there are an infinite number of ways of writing a rational number as the sum of two irrational numbers.

Let x be a rational number and y be an irrational number.Suppose their sum = z, is rational.That is x + y = zThen y = z - xThe set of rational number is closed under addition (and subtraction). Therefore, z - x is rational.Thus you have left hand side (irrational) = right hand side (rational) which is a contradiction.Therefore, by reducio ad absurdum, the supposition that z is rational is false, ie the sum of a rational and an irrational must be irrational.

Yes. The sum of two irrational numbers can be rational, or irrational.

The proposition is not true.pi and -pi are both irrational. But their sum, = 0, is rational.

The sum is a rational number.

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