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Binary System of Numbers

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Learning objective
Understand the binary number system and place values.

Understanding the Binary System of Numbers

The binary number system is a base-2 numeral system that uses only two digits: 0 and 1. It is the foundation of all modern digital computers and electronic devices because it aligns perfectly with the two-state nature of electronic circuits (off/on, false/true).

1. Basics of the Binary System

In the binary system, each digit is called a bit (short for binary digit). The position of each bit represents a power of 2, starting from the rightmost bit (least significant bit).

For example, the binary number \( 1011 \) represents:

\[ 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11 \]

This is analogous to the decimal system, which is base-10, using digits 0 through 9, where each position represents a power of 10.

2. Place Value in Binary

Each place in a binary number corresponds to an increasing power of 2 from right to left:

Bit Position 3 2 1 0
Power of 2 \(2^3 = 8\) \(2^2 = 4\) \(2^1 = 2\) \(2^0 = 1\)
Binary Digit 1 0 1 1

Thus, the value is \(1 \times 8 + 0 \times 4 + 1 \times 2 + 1 \times 1 = 11\).

3. Conversion Between Decimal and Binary

Decimal to Binary Conversion

To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. The binary number is the sequence of remainders read from bottom to top.

Example: Convert decimal 13 to binary.

  • 13 ÷ 2 = 6 remainder 1
  • 6 ÷ 2 = 3 remainder 0
  • 3 ÷ 2 = 1 remainder 1
  • 1 ÷ 2 = 0 remainder 1

Reading remainders from bottom to top: \(1101_2\).

Binary to Decimal Conversion

To convert a binary number to decimal, multiply each bit by its corresponding power of 2 and sum the results.

Example: Convert binary \(1010_2\) to decimal.

\[1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 0 + 2 + 0 = 10\]

4. Binary Arithmetic

Binary arithmetic follows simple rules similar to decimal arithmetic but limited to digits 0 and 1.

Operation Rule
Addition \(0 + 0 = 0\)
\(0 + 1 = 1\)
\(1 + 0 = 1\)
\(1 + 1 = 10\) (write 0, carry 1)
Subtraction \(0 - 0 = 0\)
\(1 - 0 = 1\)
\(1 - 1 = 0\)
\(0 - 1 = 1\) (borrow 1 from next higher bit)

Multiplication and division are performed similarly to decimal but using binary rules.

5. Applications of Binary System

  • Computing: All digital computers use binary internally to represent data and perform calculations.
  • Data Representation: Text, images, and sounds are encoded in binary for storage and transmission.
  • Logic Circuits: Binary logic forms the basis of logic gates and circuits used in electronics.

6. Connection to Sets and Subsets (Brief Overview)

Though this subtopic focuses on binary numbers, understanding binary is useful in set theory, especially in representing subsets. For example, the power set of a set with \(n\) elements has \(2^n\) subsets, analogous to the number of binary strings of length \(n\).

This connection helps solve problems related to subsets, power sets, and set operations efficiently.

Binary number: 1 0 1 1 \(2^3 = 8\) \(2^2 = 4\) \(2^1 = 2\) \(2^0 = 1\)

Worked Examples

Example 1: Convert Decimal 45 to Binary (Easy)

Solution:

  • 45 ÷ 2 = 22 remainder 1
  • 22 ÷ 2 = 11 remainder 0
  • 11 ÷ 2 = 5 remainder 1
  • 5 ÷ 2 = 2 remainder 1
  • 2 ÷ 2 = 1 remainder 0
  • 1 ÷ 2 = 0 remainder 1

Reading remainders from bottom to top: \(101101_2\).

Example 2: Convert Binary \(11010_2\) to Decimal (Easy)

Solution:

\[ 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 16 + 8 + 0 + 2 + 0 = 26 \]

Example 3: Add Binary Numbers \(1011_2\) and \(1101_2\) (Medium)

Solution:

Carry1101
First number1011
Second number1101
Sum11000

Stepwise addition:

  • Rightmost bit: \(1 + 1 = 10\) → write 0, carry 1
  • Next bit: \(1 + 1 + 1 (carry) = 11\) → write 1, carry 1
  • Next bit: \(0 + 0 + 1 (carry) = 1\) → write 1, carry 0
  • Leftmost bit: \(1 + 1 + 0 = 10\) → write 0, carry 1
  • Carry 1 is written as the new leftmost bit

Final sum: \(11000_2\).

Example 4: Find the Number of Subsets of Set \(S = {a, b, c, d}\) (Easy)

Solution:

Number of elements \(n = 4\).

Number of subsets = \(2^n = 2^4 = 16\).

Example 5: Find the Maximum Number of Elements in \(A \cap B\) if \(n(A) = 4\) and \(n(B) = 3\) (Medium)

Solution:

The maximum number of elements in the intersection is the smaller of the two set sizes.

\[ \max n(A \cap B) = \min(n(A), n(B)) = \min(4, 3) = 3 \]

Example 6: If \(n(A) = 37\), \(n(B) = 25\), and \(n(A \cup B) = 50\), find \(n(A \cap B)\) (Easy)

Solution:

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] \[ 50 = 37 + 25 - n(A \cap B) \] \[ n(A \cap B) = 37 + 25 - 50 = 12 \]

Formula Bank

  • Number of subsets of a set with \(n\) elements: \(2^n\)
  • Decimal to binary conversion: Repeated division by 2, remainders form binary digits
  • Binary to decimal conversion: \(\displaystyle \sum_{i=0}^{n-1} b_i \times 2^i\), where \(b_i\) is the bit at position \(i\)
  • Union and intersection relation: \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)
  • Maximum intersection size: \(\max n(A \cap B) = \min(n(A), n(B))\)
  • Binary addition rules: \(0+0=0\), \(0+1=1\), \(1+1=0\) (carry 1)
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