Test: Rational And Irrational Numbers - Class 9 MCQ

Dividing a rational number by zero is undefined in mathematics. This is because there is no number that can be multiplied by zero to yield a non-zero number, highlighting a fundamental rule in arithmetic.

To find an irrational number between two positive numbers a and b, you can use the formula √(ab). This method takes the geometric mean, which is often not a perfect square, resulting in an irrational number.

The product of a non-zero rational number and an irrational number is always irrational. This is because multiplying by a non-zero rational number does not change the non-terminating, non-repeating nature of the irrational number.

The average of two rational numbers is always a rational number. This is because adding two rational numbers results in a rational number, and dividing by 2 (a rational number) maintains the rationality.

To find n rational numbers between two numbers x and y, you calculate d = (y - x)/(n - 1). This method ensures that the numbers are evenly distributed between x and y, illustrating the density of rational numbers.

A rational number is defined as a number that can be expressed in the form a/b, where a and b are integers, and b is not zero. This definition includes all integers and fractions, making it a fundamental concept in understanding the number system.

Irrational numbers are defined by their non-terminating and non-repeating decimal expansions. Examples include numbers like √2 and π. Unlike rational numbers, they cannot be represented as a simple fraction, which makes them unique in the number system.

√3 is an example of an irrational number, as it cannot be expressed as a fraction of two integers and its decimal expansion is non-terminating and non-repeating. This highlights the contrast between rational and irrational numbers.

Two rational numbers a/b and c/d are equal if the cross products are equal, meaning a × d = b × c. This property allows for the comparison of fractions without converting them to a common denominator.

The rationalizing factor of the surd √2 is √2 itself, which is used to eliminate the square root from the denominator in expressions. By multiplying by √2, we can achieve a rational denominator, which is crucial in simplifying expressions in algebra.

The decimal representation of rational numbers can be either terminating or non-terminating recurring. This characteristic is crucial for distinguishing rational numbers from irrational numbers, which have non-terminating, non-repeating decimals.

To find a rational number between two positive numbers x and y, you can calculate (x + y)/2. This gives the midpoint, which is itself a rational number, illustrating the density of rational numbers on the number line.

The decimal representation of 1/8 is 0.125, which is a terminating decimal. This demonstrates how fractions can be converted to decimal form, highlighting the relationship between rational numbers and their decimal representations.

A terminating decimal is one that has a finite number of digits after the decimal point. For instance, 1/2 equals 0.5, which terminates. In contrast, 1/3 (0.333...) and 1/7 (0.142857...) are examples of non-terminating decimals.

The LCM (Least Common Multiple) method is used to convert two rational numbers to a common denominator, which makes it easier to find additional rational numbers that lie between them, ensuring no gaps in the number line.