
1. Give an example of 2 irrational numbers whose sum is rational.
2. Give an example of 2 irrational numbers whose product is rational.
Answer
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Hint: We will think of any 2 irrational numbers, we will find their sum and see if it is a rational number. We will think of any 2 irrational numbers, we will find their product and see if it is a rational number.
Formula used:
Complete step-by-step answer:
We will look at some properties of rational and irrational numbers that will help us in solving the question:
We know that a number is a rational number if it can be expressed in the form where and are integers with no common factor and .
We know that a number is an irrational number if it cannot be expressed in the form where and are integers with no common factor and .
We know that the sum of a rational and an irrational number is always an irrational number.
We know that the difference of a rational and an irrational number is always an irrational number.
We know that is an irrational number.
We can conclude from the 3rd property that is an irrational number.
We can conclude from the 4th property that is an irrational number.
We will take the first number as and the second number as .
We will find the sum of the 2 numbers:
=
= +
= 4
4 is a rational number.
We will find the product of the 2 numbers. We will substitute 2 for and for in the formula:
.
=
= 4 - 7
= - 3
is a rational number.
and are 2 irrational numbers whose sum as well as the product are rational numbers.
Note: We must know that the square roots of all non-negative integers except the perfect squares are irrational numbers. This will help us in choosing the correct irrational number. For example, etc are irrational numbers whereas etc are rational numbers.
Formula used:
Complete step-by-step answer:
We will look at some properties of rational and irrational numbers that will help us in solving the question:
We know that a number is a rational number if it can be expressed in the form
We know that a number is an irrational number if it cannot be expressed in the form
We know that the sum of a rational and an irrational number is always an irrational number.
We know that the difference of a rational and an irrational number is always an irrational number.
We know that
We can conclude from the 3rd property that
We can conclude from the 4th property that
We will take the first number as
We will find the sum of the 2 numbers:
=
=
= 4
4 is a rational number.
We will find the product of the 2 numbers. We will substitute 2 for
=
= 4 - 7
= - 3
Note: We must know that the square roots of all non-negative integers except the perfect squares are irrational numbers. This will help us in choosing the correct irrational number. For example,
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