JEE Mathematics Number Systems Notes Set 01

Place Value and Base

In every positional number system the value of a digit depends on where it stands. The number of different symbols the system uses is called its base (or radix). The denary or decimal system has base 10, the binary system has base 2, the ternary system has base 3, the quinary system has base 5, the octal system has base 8 and the hexadecimal system has base 16. A small subscript written after the numeral shows the base, for example \( (34)_{5} \) is a quinary numeral and \( (25)_{8} \) is an octal numeral. The numeral \( (34)_{5} \) is read as three four to the base five, not as thirty-four.

Rule: in base b the digits used are 0, 1, ..., b - 1, and the digit in the k-th place from the right (counting the units place as the 0-th) is multiplied by \( b^k \). A digit equal to or larger than the base is never allowed, so 5 can never appear in a quinary numeral.

Computers store information as the presence or absence of an electrical signal, so they naturally use the binary system. Octal and hexadecimal are used as short ways of writing long binary numbers.

Converting to the Denary System

Write the numeral in expanded form with place values written in denary, then add. Always do the arithmetic in base 10.

The same digits give different values in different bases. For the digits 1111: in base 10 the value is 1111, in base 2 it is 15, in base 5 it is 156 and in base 8 it is 585.

Converting from the Denary System

Divide the number by the new base, write down the remainder, and keep dividing the quotient until the quotient is smaller than the base. The remainders read from the last to the first form the answer.

Example: convert 29 to base 2.
29 = 2 x 14 remainder 1
14 = 2 x 7 remainder 0
7 = 2 x 3 remainder 1
3 = 2 x 1 remainder 1
1 = 2 x 0 remainder 1
Reading the remainders upwards: 29 = \( (11101)_{2} \).

Example: convert 487 to base 5.
487 = 5 x 97 remainder 2
97 = 5 x 19 remainder 2
19 = 5 x 3 remainder 4
3 = 5 x 0 remainder 3
Reading the remainders upwards: 487 = \( (3422)_{5} \).

Example: convert 526 to base 3.
526 = 3 x 175 remainder 1
175 = 3 x 58 remainder 1
58 = 3 x 19 remainder 1
19 = 3 x 6 remainder 1
6 = 3 x 2 remainder 0
2 = 3 x 0 remainder 2
Reading the remainders upwards: 526 = \( (201111)_{3} \).

Converting Between Two Bases Other Than 10

Convert first to denary and then from denary to the required base.

Example: convert \( (340)_{5} \) to base 3.
\( (340)_{5} \) = 3 x 25 + 4 x 5 + 0 = 95.
Now 95 in base 3 is \( (10112)_{3} \).

Next Number in a Base

To find the successor, add 1 to the units digit. If the digit would become equal to the base, write 0 and carry 1 to the next place. The successor of the largest four-digit numeral \( (4444)_{5} \) is therefore \( (10000)_{5} \), which is \( 5^4=625 \) in denary.

Addition and Subtraction in Another Base

Add column by column from the right. When a column total reaches the base or more, write down the remainder after dividing by the base and carry the quotient. In subtraction, when the upper digit is smaller, borrow one unit from the next place; that unit is worth the base, not 10.

Example (addition): \( (323)_{5} \) + \( (443)_{5} \). Units: 3 + 3 = 6 = 1 x 5 + 1, write 1 and carry 1. Fives: 2 + 4 + 1 = 7 = 1 x 5 + 2, write 2 and carry 1. Twenty-fives: 3 + 4 + 1 = 8 = 1 x 5 + 3, write 3 and carry 1. The carry gives a new leading 1. Result: \( (1321)_{5} \) (check: 88 + 123 = 211).

Example (subtraction): \( (2222)_{5} \) - \( (423)_{5} \). Units: 2 - 3 is not possible, borrow 5, so 7 - 3 = 4. Fives: the 2 became 1, so 1 - 2 is not possible, borrow 5, so 6 - 2 = 4. Twenty-fives: the 2 became 1, so 1 - 4 is not possible, borrow 5, so 6 - 4 = 2. Hundred-twenty-fives: the 2 became 1, so 1 - 0 = 1. Result: \( (1244)_{5} \) (check: 312 - 113 = 199).

Multiplication in Another Base

The easiest method is to convert both numbers to denary, multiply, and convert the product back to the required base. The two numbers may even be in different bases.

Example: \( (201)_{5} \) x \( (130)_{5} \). \( (201)_{5} \) = 51 and \( (130)_{5} \) = 40. The product is 2040, which in base 5 is \( (31130)_{5} \).

Example (different bases): \( (111)_{2} \) x \( (221)_{3} \). \( (111)_{2} \) = 7 and \( (221)_{3} \) = 25, so the product is 175. In binary 175 = \( (10101111)_{2} \) and in ternary 175 = \( (20111)_{3} \).

Quick Reference

1. The base equals the number of distinct symbols; the largest digit is one less than the base.
2. Value of a numeral = sum of (digit x base to the power of its place number).
3. To go from denary to base b, divide repeatedly by b and read the remainders upwards.
4. For two non-denary bases, go through denary.
5. Hexadecimal digits: A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
6. A numeral with a digit not less than its base is invalid.
7. The successor of the largest n-digit numeral in base b is 1 followed by n zeros.