Number System - 1

To convert hexadecimal to binary, replace each hex digit with its 4-bit binary equivalent: 1=0001, A=1010, 0=0000, 3=0011. Therefore (1A03)₁₆ = 0001 1010 0000 0011₂.

First convert octal to binary: (765)₈ = 111 110 101₂ = 111110101₂. Then group binary digits into 4-bit sets from right: 1 1111 0101. Convert to hex: 1=1, 1111=F, 0101=5, giving (1F5)₁₆.

To convert binary to octal, group the binary digits into sets of 3 from right: 111 010₂. Convert each 3-bit group: 111₍₂₎ = 7₍₈₎, 010₍₂₎ = 2₍₈₎. Therefore 111010₂ = 72₈.

To convert decimal to hexadecimal, divide by 16 repeatedly: 316 ÷ 16 = 19 remainder 12 (C), 19 ÷ 16 = 1 remainder 3, 1 ÷ 16 = 0 remainder 1. Reading remainders from last to first: (13C)₁₆.

Computers perform all four basic mathematical operations. While addition is the fundamental operation at the hardware level (subtraction via 2's complement, multiplication via repeated addition, division via repeated subtraction), all operations are implemented at the instruction set level.

To represent 39 in 2's complement: first convert 39 to binary (100111), then take 1's complement (011000), add 1 (011001). For 8-bit representation, this is 00111001. However, since 39 is positive, its 2's complement form is simply its binary representation 00111001, which when written without leading zeros is 11011001.

Floating point notation normalizes a binary number by expressing it as (mantissa) × 2^(exponent). For (1101.111)₂, normalize by moving the decimal point 4 places left: 0.1101111 × 2⁴, where the mantissa is 0.1101111 and exponent is 4.

In any number system with radix r, after exhausting all single digits (0 to V, where V is the (r-1)th digit), the next number is written as 10 (meaning 1×r¹ + 0×r⁰). In base-32, 'V' is the 31st digit (index 30), so 'V' is followed by (10)₃₂.

To convert binary (1110101)₂ to decimal, sum powers of 2: 1×2⁶ + 1×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 0×2¹ + 1×2⁰ = 64 + 32 + 16 + 0 + 4 + 0 + 1 = 117.

In hexadecimal, adding 1 to FFEF gives FFF0. Subtracting 1 from FFEF gives FFEE because the last digit F (value 15) becomes E (value 14).

1101 binary equals 13 decimal. 101 binary equals 5 decimal. Their sum 18 in decimal converts to 10010 in binary.

128 divided by 32 gives quotient 4 with remainder 0. In positional notation, this becomes (40)32 where 4 is in the 32s place and 0 is in the units place.

First we find out 9’s complement as follows:
99999 - 56789 = 43210
Then we add, 43210 + 1 = 43211.

When the base is 7, then number of 4-digits numbers which can be represented is 7⁴ = 2401 with a range from 0000-2400. Hence, 2400 is the right answer.

1010 is a forbidden code in the above series.