1.7 Number Bases

Number bases represent the numerical system in which values are expressed. The base indicates the number of unique digits, including zero, that a numeral system uses. Common bases include:

• Base 10 (Decimal):

  Standard numeric system.

• Base 2 (Binary):

  Used in computer systems.

• Base 16 (Hexadecimal):

  Common in programming and computing.

Key Concepts:

• Converting numbers between bases involves successive division by the new base and noting remainders.
• To perform operations on numbers in any base, first convert them to base 10, perform the operation, and then convert the answer back to the original base.

Example 1: Conversion from Decimal to Binary

Convert \(18\) from base 10 to base 2.

Solution:

Step 1: Divide by 2 and record remainders: \[ 18 \div 2 = 9 \text{ remainder } 0 \] \[ 9 \div 2 = 4 \text{ remainder } 1 \] \[ 4 \div 2 = 2 \text{ remainder } 0 \] \[ 2 \div 2 = 1 \text{ remainder } 0 \] \[ 1 \div 2 = 0 \text{ remainder } 1 \] Step 2: Write the remainders in reverse order: \[ 18_{10} = 10010_2 \]

Example 2: Conversion from Binary to Decimal

Convert \(1011_2\) to base 10.

Solution:

Apply place values: \[ (1 \cdot 2^3) + (0 \cdot 2^2) + (1 \cdot 2^1) + (1 \cdot 2^0) = 8 + 0 + 2 + 1 = 11 \] Therefore, \(1011_2 = 11_{10}\).

Example 3: Multiplication in Base 3

Simplify \(21_3 \cdot 2_3\) in base 3.

Solution:

Convert each to decimal: \[ 21_3 = (2 \cdot 3^1) + (1 \cdot 3^0) = 6 + 1 = 7, \quad 2_3 = 2 \] Multiply in decimal: \[ 7 \cdot 2 = 14 \] Convert 14 to base 3: \[ 14 \div 3 = 4 \text{ remainder } 2, \quad 4 \div 3 = 1 \text{ remainder } 1, \quad 1 \div 3 = 0 \text{ remainder } 1 \] Write the remainders in reverse: \[ 14_{10} = 112_3 \]

Example 4: Squaring in Base 2

Simplify \((11_{\text{two}})^2\).

Solution:

Convert to decimal: \[ 11_{\text{two}} = 3 \] Square in decimal: \[ 3^2 = 9 \] Convert 9 back to binary: \[ 9 \div 2 = 4 \text{ remainder } 1 \] \[ 4 \div 2 = 2 \text{ remainder } 0 \] \[ 2 \div 2 = 1 \text{ remainder } 0 \] \[ 1 \div 2 = 0 \text{ remainder } 1 \] Write in reverse order: \[ 9_{10} = 1001_2 \]