Test: Number Systems - 1 - Grade 9 MCQ

Octal to Decimal conversion is obtained by multiplying 8 to the power of base index along with the value at that index position.

(532.2)8 = 5 * 8² + 3 * 8¹ + 2 * 8⁰ + 2 * 8^-1 = (346.25)10

Factoring √12 and √27 and solving, we get
√12 = √(4 x 3) = √(2 x 2 x 3) = 2√3
√27 = √(9 x 3) = √(3 x 3 x 3) = 3√3

Now the equation we get is 3√12 / 6√27. On substituting the values from above, we get

- A rational number is a number that can be expressed as a fraction where both the numerator and denominator are integers.
- √196 is a rational number because it simplifies to 14/1, which is a fraction where both the numerator and denominator are integers.
- Options B and C are irrational numbers because they cannot be expressed as fractions.
- Option A is a repeating decimal, which can be rational if it eventually settles into a repeating pattern, but without further information, it is not clear if this is the case.

And we know that the value of 11√2 is greater than 15 so it's value will be positive, And also sum or differences of rational and irrational is irrational

Every rational number is a real number. Real Number is a set of numbers formed by both Rational and Irrational numbers are combined.

A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly.

Decimals of this type cannot be represented as fractions, and as a result, are irrational numbers.

An irrational number is a type of number that cannot be expressed as a simple fraction. Its decimal representation has unique characteristics:

• Non-terminating: The decimal goes on forever without ending.
• Non-repeating: There are no repeating patterns in the digits.

Examples of irrational numbers include:

• Pi (π): Approximately 3.14159, it continues infinitely without repetition.
• The square root of 2: Approximately 1.41421, also non-terminating and non-repeating.

In summary, the decimal representation of an irrational number is neither terminating nor repeating.

A number that cannot be expressed as a terminating or repeating decimal is classified as an irrational number. Here are some key points to understand this concept:

• Rational numbers can be written as fractions, which either terminate (like 0.5) or repeat (like 0.333...).
• Irrational numbers, on the other hand, cannot be written as simple fractions. Their decimal representations are non-terminating and non-repeating.
• Examples of irrational numbers include π (pi) and √2 (the square root of 2).
• Irrational numbers are essential in mathematics, particularly in geometry and calculus.

In summary, a number that cannot be expressed as either a terminating or a repeating decimal is indeed called an irrational number.

Zero (0) is an integer, and all integers are rational numbers because they can be expressed as a fraction where the denominator is 1 (e.g., 0/1).

The other options involve irrational numbers that cannot be expressed as simple fractions. Thus, 0 is the only rational number among the given choices.

The given expression is in the form of (a - b)(a + b), which is a standard identity:

(a - b)(a + b) = a² - b²

Here,
a = 3
b = √3

Now apply the identity:

(3 - √3)(3 + √3) = 3² - (√3)² = 9 - 3 = 6

So, the result is 6, which is a rational number.

We use the identity
(a + b)² = a² + 2ab + b²

Here,
a = √5
b = √7

Now apply the identity:

(√5 + √7)² = (√5)² + 2 × √5 × √7 + (√7)²
= 5 + 2√35 + 7
= 12 + 2√35