Which of the following statements is false?
The sum of two rational numbers is always rational.
The sum of a rational number and an irrational number is always rational.
The product of a nonzero rational number and an irrational number is always irrational.
The product of two irrational numbers is either rational or irrational.
"The sum of two rational numbers is always rational." - true, for example, 2+0.5=2.5 (decimals and fractions are rational)
So statement is true.
"The product of a nonzero rational number and an irrational number is always irrational." - true, 5*√2=5(√2) remains irrational. But watch 0*√2=0 (rational). However, 0 is not "nonzero", so the statement remains true.
"The product of two irrational numbers is either rational or irrational." √2*√3=√6 - irrational
√2*√2=√4=2 - rational.
So statement is true.
Finally,
"The sum of a rational number and an irrational number is always rational."
I can easily find examples where the sum of rational and irrational is irrational, as in:
2+√2 : irrational
but unable to find a case where the sum is rational.
Since the statement says "always rational", one single counter-example (as I gave above) invalidates the statement.
so the answer is D?
The false statement among the options is: "The product of a nonzero rational number and an irrational number is always irrational."
To determine which of the statements is false, let's analyze each statement one by one:
1. The sum of two rational numbers is always rational.
To verify this statement, we need to understand that the sum of rational numbers is calculated by adding their numerators and retaining the common denominator. Since the addition of rational numbers follows the same rules as integer addition, the sum of two rational numbers will always be rational. Hence, this statement is true.
2. The sum of a rational number and an irrational number is always rational.
In this statement, we are dealing with the addition of a rational number and an irrational number. It is important to note that irrational numbers cannot be expressed as fractions or ratios of integers. Since rational numbers can be expressed as fractions, their sum with an irrational number cannot be simplified into a rational form. Therefore, the sum of a rational number and an irrational number is always irrational. Thus, this statement is false.
3. The product of a nonzero rational number and an irrational number is always irrational.
To assess this statement, we consider that multiplying an irrational number by a nonzero rational number will still yield an irrational number. This holds true because the product of any nonzero rational number and an irrational number cannot be expressed as a ratio of integers, hence making it an irrational number. Therefore, this statement is true.
4. The product of two irrational numbers is either rational or irrational.
In this statement, we are examining the product of two irrational numbers. It is important to note that the product of irrational numbers may yield a rational or an irrational number. For example, multiplying the irrational number √2 by itself results in a rational number (2). However, multiplying two different irrational numbers such as √2 and √3 will result in an irrational number (√6). Therefore, this statement is true.
To summarize, the false statement is the second one: "The sum of a rational number and an irrational number is always rational."