2.1 Number Bases
Introduction to Number Bases
A number base is a system of counting that uses a specific set of digits to represent numbers. The digits used in a base $n$ system range from 0 to $n - 1$.
Digits Used in Various Number Bases
- Base 2 (Binary): 0, 1
- Base 3: 0, 1, 2
- Base 4: 0, 1, 2, 3
- Base 5: 0, 1, 2, 3, 4
- Base 6: 0, 1, 2, 3, 4, 5
- Base 7: 0, 1, 2, 3, 4, 5, 6
- Base 8 (Octal): 0, 1, 2, 3, 4, 5, 6, 7
- Base 9: 0, 1, 2, 3, 4, 5, 6, 7, 8
- Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Place Value and Digit Value
In a number expressed in base $n$, the place value of each digit from right to left is represented by powers of $n$ ($n^0, n^1, n^2, n^3, \dots$).
The digit value is calculated by multiplying the digit by its corresponding place value:
$$\text{Digit Value} = \text{Digit} \times \text{Place Value}$$
Numerical Value of a Number in Base $n$
The total value of a number in base 10 is found by summing the products of each digit and its place value.
Example: Express $342_5$ in Base 10:
$$342_5 = (3 \times 5^2) + (4 \times 5^1) + (2 \times 5^0) = 75 + 20 + 2 = 97_{10}$$
2.2 Conversion of Number Bases
1. Converting Base $n$ to Base 10
Expand the number using place values and calculate the sum.
2. Converting Base 10 to Base $n$
Use the repeated division method by dividing the Base 10 number by $n$. Write down the remainder at each step until the quotient becomes zero. The converted number is read from the bottom remainder to the top remainder.
3. Converting Base $p$ to Base $q$ (where $p, q \neq 10$)
- Convert Base $p$ to Base 10 using place value expansion.
- Convert the resulting Base 10 number to Base $q$ using repeated division.
4. Direct Conversion Between Base 2 and Base 8
- Base 2 to Base 8: Group the binary digits into sets of 3 digits from right to left, then convert each group into its single-digit octal equivalent.
- Base 8 to Base 2: Convert each octal digit directly into a 3-digit binary representation.
2.3 Calculations Involving Number Bases
Addition and Subtraction of Numbers in various Bases
Calculations in base $n$ follow the standard column-based vertical format with special base-borrowing rules:
- Addition: When the sum of digits in a column reaches or exceeds the base $n$, divide the sum by $n$. Write down the remainder as the column answer and carry over the quotient to the next column.
- Subtraction: When borrowing from the left column, the borrowed value equals the base $n$ (instead of 10).