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Statement (A): Between two real number, there is an integer.
Statement (B): Between two real number, there is a rational number.
Statement (B): Between two real number, there is a rational number.
- a)Both statements A and B are true.
- b)Both statements A and B are false.
- c)A is true but R is false
- d)None of the above
Correct answer is option 'D'. Can you explain this answer?
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Statement (A): Between two real number, there is an integer.Statement ...
Statement A is wrong but statement B is right ..because integer is uncountable there exist (-infinity to +infinity) ..in between two integer no. real no. will exists
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Statement (A): Between two real number, there is an integer.Statement ...
Statement (A): Between two real numbers, there is an integer.
This statement is false. In general, there are infinitely many real numbers between any two given real numbers, and not all of them are integers. For example, between 1 and 2, we have numbers like 1.1, 1.5, 1.9, etc. None of these numbers are integers.
Statement (B): Between two real numbers, there is a rational number.
This statement is true. The set of rational numbers includes all numbers that can be expressed as a fraction of two integers. Since there are infinitely many rational numbers, we can always find a rational number between any two given real numbers.
Explanation:
To understand the difference between these two statements, it is important to understand the properties of different number systems.
- Real Numbers: Real numbers include all rational and irrational numbers. They can be represented on a number line and occupy the entire number line. Examples of real numbers include integers, fractions, decimals, square roots, pi, etc.
- Integers: Integers are a subset of real numbers. They include all positive and negative whole numbers (including zero) but do not include fractions or decimals.
- Rational Numbers: Rational numbers are also a subset of real numbers. They include all numbers that can be expressed as a fraction of two integers. Rational numbers can be written as terminating or repeating decimals.
Now, let's consider an example to illustrate the difference between the two statements:
Example: Consider the two real numbers 1 and 2.
- Statement (A) claims that there is an integer between 1 and 2. However, this is not true since the only integers between 1 and 2 are 1 and 2 themselves, and they are not between the two numbers.
- Statement (B) claims that there is a rational number between 1 and 2. This is true because there are infinitely many rational numbers between any two given real numbers. For example, we can choose the rational number 1.5, which lies between 1 and 2.
Therefore, the correct answer is option D: None of the above.
This statement is false. In general, there are infinitely many real numbers between any two given real numbers, and not all of them are integers. For example, between 1 and 2, we have numbers like 1.1, 1.5, 1.9, etc. None of these numbers are integers.
Statement (B): Between two real numbers, there is a rational number.
This statement is true. The set of rational numbers includes all numbers that can be expressed as a fraction of two integers. Since there are infinitely many rational numbers, we can always find a rational number between any two given real numbers.
Explanation:
To understand the difference between these two statements, it is important to understand the properties of different number systems.
- Real Numbers: Real numbers include all rational and irrational numbers. They can be represented on a number line and occupy the entire number line. Examples of real numbers include integers, fractions, decimals, square roots, pi, etc.
- Integers: Integers are a subset of real numbers. They include all positive and negative whole numbers (including zero) but do not include fractions or decimals.
- Rational Numbers: Rational numbers are also a subset of real numbers. They include all numbers that can be expressed as a fraction of two integers. Rational numbers can be written as terminating or repeating decimals.
Now, let's consider an example to illustrate the difference between the two statements:
Example: Consider the two real numbers 1 and 2.
- Statement (A) claims that there is an integer between 1 and 2. However, this is not true since the only integers between 1 and 2 are 1 and 2 themselves, and they are not between the two numbers.
- Statement (B) claims that there is a rational number between 1 and 2. This is true because there are infinitely many rational numbers between any two given real numbers. For example, we can choose the rational number 1.5, which lies between 1 and 2.
Therefore, the correct answer is option D: None of the above.
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