Number System 2

V. Converting Numbers from One Base to Another

A) Converting Any Base to Decimal

Method:

1. Find the position (column) value of each digit
2. Multiply each digit by its position value
3. Add up all the products

B) Converting Decimal to Any Base — Division-Remainder Method

Method:

1. Divide the decimal number by the new base
2. Write down the remainder — this is the rightmost (last) digit of the answer
3. Divide the quotient again by the same base
4. Repeat this process until the quotient becomes 0
5. Read all the remainders from BOTTOM to TOP — that gives the final answer

C) Converting One Base to Another Base (Not Decimal)

Method:

1. First convert the given number to decimal
2. Then convert that decimal number to the target base

VI. Shortcut Conversion Tricks

Because Binary, Octal and Hexadecimal are all closely related (8 = 2³ and 16 = 2⁴), we can convert between them quickly — WITHOUT going through decimal.

A) Binary → Octal

1. Starting from the right, group the binary digits into sets of 3
2. Convert each group of 3 bits into its equivalent octal digit

B) Octal → Binary

1. Convert each octal digit into a 3-bit binary group
2. Join all the groups together

C) Binary → Hexadecimal

1. Starting from the right, group the binary digits into sets of 4
2. Convert each group of 4 bits into its equivalent hex digit

D) Hexadecimal → Binary

1. Convert each hex digit into a 4-bit binary group
2. Join all the groups together

VII. Fractional Numbers

Just like the decimal system has digits after the decimal point (e.g. 3.14), other number systems can also have digits after their own 'point' (binary point, octal point, etc.).

Rule:

• Digits to the LEFT of the point use positive powers of the base: base⁰, base¹, base², …
• Digits to the RIGHT of the point use negative powers of the base: base⁻¹, base⁻², base⁻³, …