Base System Concepts and Questions for CAT

Base System Concepts and Questions for CAT– We humans commonly use the decimal system (base 10) for our mathematical calculations. However, computers work on the binary system (base 2), and some of us from engineering or science backgrounds are also familiar with octal (base 8) and hexadecimal (base 16) systems. But the base can be any number, depending on the number of different digits used in that system.

For example, in the decimal system, the base is 10, and we use the digits 0-9. In base 2, we use 0 and 1 as the digits, and in the hexadecimal system, with a base of 16, we use the digits 0-9 and the alphabets A-F (representing values from 10 to 15) to have 16 different digits. A new base, such as 5, would use the digits 0-4.

The problems in competitive exams related to base systems are based on two concepts: conversion of a number from one base to another, and simple mathematical operations (addition, subtraction, and multiplication) of numbers in bases other than 10.

To convert a number from base 10 to another base:

1. Divide the number by the desired base.
2. The remainder becomes the unit digit in the new base, and the quotient is further divided.
3. Repeat the process until the quotient becomes zero.

For example, to convert 435 into base 6:
– Divide 435 by 6: quotient = 72, remainder = 3. So, the unit digit in base 6 is 3.
– Divide 72 by 6: quotient = 12, remainder = 0. So, the ten’s digit in base 6 is 0.

Therefore, 435 in base 10 = 2003 in base 6.

To convert a number from another base to base 10:

1. Multiply the digits of the number by the corresponding powers of the base.
2. Add the results.

For example, to convert (3564)7 into base 10:
– (3 x 7^3) + (5 x 7^2) + (6 x 7^1) + (4 x 7^0) = (1230)10.

To convert a decimal/fraction into another base:

1. Multiply the decimal by the base.
2. Remove the whole number part after the multiplication, which becomes the first digit after the decimal in the new base.
3. Repeat step 2 until all numbers after the decimal are exhausted.

For example, to convert 0.256 from base 10 to base 5:
– (0.256 x 5) = 1.28. The whole number part is 1.
– (0.28 x 5) = 1.4. The whole number part is 1.
– (0.4 x 5) = 2.0. The whole number part is 2.

Therefore, (0.256)10 = (0.112)5.

To convert a fraction from another base back to decimal:

For a number (0.abc) in base x, the decimal equivalent is:
– (a x (1/x)) + (b x (1/x^2)) + (c x (1/x^3)).

For example, to convert (0.112)5 back to decimal:
– (1 x (1/5)) + (1 x (1/5^2)) + (2 x (1/5^3)) = (0.256)10.

The process of addition, subtraction, and multiplication in bases other than 10 is similar to that in decimal system, with carry over and remainder. If the sum is greater than the base, divide the sum by the base and keep the remainder, carry over the quotient. Base System Concepts and Questions for CAT.

Converting a number from base x to base y (where neither x nor y is equal to 10) can be done by converting the number to base 10 first, and then converting from base 10 to the desired base. If the base y is a power of x, the conversion is easier.

For example, to convert (10011110101)2 to base 4:
– Convert the digits used in base 4 to base 2: (0)4 = (00)2, (1)4 = (01)2, (2)4 = (10)2, (3)4 = (11)2.
– Convert each digit of the given number to base 2 using the above conversions: (10011110101)2 = (10110101)2.
– Convert the result from base 2 to base 4 using the digit conversions: (10110101)2 = (255)4.

These concepts and techniques are used in solving problems involving different base systems in competitive exams.