Basics of Number System Video Lecture | General Aptitude for GATE - Mechanical Engineering

Ans. The most common types of number systems are: 1. <b>Natural Numbers (N)</b>: The set of positive integers starting from 1 (1, 2, 3, ...). 2. <b>Whole Numbers (W)</b>: Similar to natural numbers but include 0 (0, 1, 2, 3, ...). 3. <b>Integers (Z)</b>: The set of whole numbers and their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...). 4. <b>Rational Numbers (Q)</b>: Numbers that can be expressed as the quotient of two integers (a/b, where b ≠ 0). 5. <b>Irrational Numbers</b>: Numbers that cannot be expressed as a simple fraction (e.g., √2, π). 6. <b>Real Numbers (R)</b>: All rational and irrational numbers combined. 7. <b>Complex Numbers (C)</b>: Numbers that include a real part and an imaginary part (a + bi, where i is the imaginary unit).

Ans. To convert a decimal number to a binary number, follow these steps: 1. Divide the decimal number by 2. 2. Record the quotient and the remainder. 3. Repeat the process with the quotient until it becomes 0. 4. The binary number is formed by reading the remainders in reverse order. For example, to convert 13 to binary: - 13 ÷ 2 = 6, remainder 1 - 6 ÷ 2 = 3, remainder 0 - 3 ÷ 2 = 1, remainder 1 - 1 ÷ 2 = 0, remainder 1 Reading the remainders from bottom to top gives 1101, so 13 in binary is 1101.

Ans. The base of a number system determines the number of unique digits used to represent numbers. For example: - In the decimal system (base 10), digits range from 0 to 9. - In the binary system (base 2), digits are limited to 0 and 1. - In the hexadecimal system (base 16), digits range from 0 to 9 and include A to F (representing 10 to 15). The base affects how numbers are represented and calculated within that system, influencing operations like addition, subtraction, and multiplication.

Ans. Arithmetic operations in different number systems follow similar principles but require careful attention to the base. - <b>Addition</b>: Align the numbers by their least significant digit, add each column, and carry over if the sum exceeds the base. - <b>Subtraction</b>: Similar to addition, but you may need to borrow from the next column if the top digit is smaller than the bottom digit. - <b>Multiplication</b>: Multiply each digit of one number by each digit of the other, accounting for the base, and then add the results. - <b>Division</b>: Similar to long division in decimal, but you need to keep track of the base during the process.

Ans. Common mistakes include: 1. <b>Confusing bases</b>: Mixing up the base of the number being worked with can lead to incorrect answers. 2. <b>Miscalculating carries or borrows</b>: When performing arithmetic operations, ensure that carries and borrows are correctly handled according to the base. 3. <b>Ignoring place value</b>: Each digit in a number has a different value depending on its position and the base; ignoring this can lead to errors. 4. <b>Not converting correctly</b>: When converting between number systems, double-check the process to avoid miscalculations. 5. <b>Rushing through calculations</b>: Take your time to ensure accuracy, especially with binary and hexadecimal operations, as they can be tricky.