Number Systems Basics

Study Notes

A number system represents numbers using digits or symbols consistently.
The value of a digit is determined by its position and the base of the number system.
Number systems facilitate conversions between different representations.

Binary: Base 2, uses digits 0 and 1.
Octal: Base 8, uses digits 0-7. Advantage: fewer digits than other systems, reducing computational errors.
Decimal: Base 10, uses digits 0-9. This is the system we typically use in everyday life.
Hexadecimal: Base 16, uses digits 0-9 and A-F.

Step 1: Determine the base of the given binary number, which is always 2.
Step 2: Multiply each digit of the binary number by the appropriate power of 2, starting from the rightmost digit with 2^0 and increasing by 1 for each subsequent digit to the left.
Step 3: Add the products calculated in step 2 to obtain the decimal equivalent.
Example:* (100111)₂
(1 x 2⁵) + (0 x 2⁴) + (0 x 2³) + (1 x 2²) + (1 x 2¹) + (1 x 2⁰)
(1 x 32) + (0 x 16) + (0 x 8) + (1 x 4) + (1 x 2) + (1 x 1)
32 + 0 + 0 + 4 + 2 + 1 = 39
Therefore, (100111)₂ is equal to (39)₁₀.

Step 1: Determine the base of the required binary number, which is always 2.
Step 2: Divide the given decimal number by the base (2).
Step 3: Record the quotient and remainder.
Step 4: Repeat steps 2 and 3 using the quotient as the new dividend until the quotient becomes less than the base (2).
Step 5: The remainders, read from bottom to top, form the binary representation of the decimal number.
Example:* Convert (17)₁₀ to binary.
17 / 2 = 8 remainder 1
8 / 2 = 4 remainder 0
4 / 2 = 2 remainder 0
2 / 2 = 1 remainder 0
1 / 2 = 0 remainder 1
Reading the remainders from bottom to top, the binary equivalent is (10001)₂.

Binary addition is the foundation for binary subtraction, multiplication, and division.
There are four basic rules for adding binary digits:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (with a carry-over of 1 to the next digit)