Binary System

Fundamental Concepts of Binary

• Binary is the primary language used by computers to communicate and process information.
• It is a numerical system that uses only two digits: $0$ and $1$.
• These digits are used to represent all numerical values and characters within a computer system.

The Decimal System (Base 10)

• In everyday life, we use the decimal system, which is based on the number $10$.
• This system utilizes ten digits: $0, 1, 2, 3, 4, 5, 6, 7, 8, \text{ and } 9$.
• Positional Value (HTU): The value of a digit is determined by its position.
  • H (Hundreds): The third position from the right.
  • T (Tens): The second position from the right.
  • U (Units/Ones): The first position from the right.
• Example: In the number $569$, the digit $5$ actually represents five hundreds ($500$) because there are two whole numbers to its right.
• Base 10 Logic: Each position to the left is worth $10 \times$ more than the place to its immediate right.
  • $100 \leftarrow 10 \leftarrow 1$
  • The progression follows: $1 \times 10 = 10$, and $10 \times 10 = 100$.

The Binary System (Base 2)

• Binary is known as 'Base 2' because it utilizes two digits.
• Base 2 Logic: Each position to the left is worth $2 \times$ more than the place to its immediate right.
• The column values increase by powers of two:
  • $16, 8, 4, 2, 1$
  • The progression follows: $1 \times 2 = 2$, $2 \times 2 = 4$, $4 \times 2 = 8$, and $8 \times 2 = 16$.
• While manually identifying columns (like HTU in decimal), one should remember these values ($1, 2, 4, 8, 16 \dots$) to perform conversions efficiently.

Converting Decimal to Binary

• Writing numbers in binary involves identifying which column values, when added together, equal the desired decimal number.

• Binary Rule: You may only use the digits $0$ (off/unused) and $1$ (on/used).

• Demonstration with Binary-Bot: To write the number $9$ as a binary number:

  • Look at the columns: $16, 8, 4, 2, 1$.
  • The highest value within $9$ is $8$, so put a $1$ in the $8$ column.
  • There is $1$ remaining ($9 - 8 = 1$).
  • Put a $0$ in the $4$ column and the $2$ column.
  • Put a $1$ in the $1$ column.
  • Calculation: $(1 \times 8) + (1 \times 1) = 9$.
  • Result: $1001$.

• Prohibited Methods: One cannot represent $9$ using other digits or improper formatting (as queried by the interlocutor):

  • You cannot use a $4$ in the column (e.g., placing the digit $4$ and a $1$ in the table) or use multiple instances of a number like $(1 \times 4) + (2 \times 2) + (1 \times 1)$ because binary only allows the digits $0$ and $1$.

• Further Examples of Conversions:

  • 10: $32(0), 16(0), 8(1), 4(0), 2(1), 1(0) \rightarrow 1010$
  • 18: $32(0), 16(1), 8(0), 4(0), 2(1), 1(0) \rightarrow 10010$
  • 33: $32(1), 16(0), 8(0), 4(0), 2(0), 1(1) \rightarrow 100001$

Binary Addition Rules and Operations

• Binary addition follows specific rules because only two digits are available ($0$ and $1$). This differs from standard decimal addition.

• Operation Rules:

  • $0 + 0 = 0$
  • $0 + 1 = 1$
  • $1 + 0 = 1$
  • $1 + 1 = 0 \text{ (with } 1 \text{ carried over to the next higher column)}$
  • $1 + 1 + 1 = 1 \text{ (with } 1 \text{ carried over to the next higher column)}$

• Worked Example Calculation:

  • Term 1: $01011101$
  • Term 2: $00110010$
  • Sum: $10001111$

Exercises and Odd Number Identification

• To determine if a binary number is odd, look at the '1's column (the far-right bit). If the far-right bit is a $1$, the number is odd; if it is a $0$, the number is even.
• Practice Conversions (Question to Answers):
  • $10111 = 11$
  • $10001 = 17$
  • $10110 = 22$
  • $10 = 2$
  • $111111 = 63$
  • $101010 = 42$
  • $10111 = 23$

ASCII (American Standard Code for Information Interchange)

• Computers use the ASCII table to map digital numbers to human-readable characters like letters and symbols.

• Alphabet ASCII Subset (Uppercase):

  • $65 = A$
  • $66 = B$
  • $67 = C$
  • $68 = D$
  • $69 = E$
  • $70 = F$
  • $71 = G$
  • $72 = H$
  • $73 = I$
  • $74 = J$
  • $75 = K$
  • $76 = L$
  • $77 = M$
  • $78 = N$
  • $79 = O$
  • $80 = P$
  • $81 = Q$
  • $82 = R$
  • $83 = S$
  • $84 = T$
  • $85 = U$
  • $86 = V$
  • $87 = W$
  • $88 = X$
  • $89 = Y$
  • $90 = Z$

• Miscellaneous ASCII Codes (Selected):

  • Digits: $48 = 0$, $49 = 1$, $50 = 2$, $51 = 3$, $52 = 4$, $53 = 5$, $54 = 6$, $55 = 7$, $56 = 8$, $57 = 9$
  • Lowercase: $97 = a$, $98 = b$, $99 = c$, $100 = d \dots 122 = z$
  • Symbols: $33 = !$, $34 = "$, $35 = \#$, 36 = \, $37 = \%$, $38 = \&$, $40 = ($, $41 = )$, $43 = +$, $44 = ,$, $45 = -$, $46 = .$, $47 = /$, $58 = :$, $59 = ;$, $60 = <$, $61 = =$, $62 = >$, $63 = ?$, $64 = @$, $91 = [$, $92 = \backslash$, $93 = ]$, $123 = \{$, $124 = |$, $125 = \}$, $127 = \text{DEL}$.

Application: Encoding Names in Binary

• To translate a name like Erick Singh, you must find the decimal value for each character in the ASCII table and convert that decimal into binary.

• Name Breakdown:

  • E: Decimal $69$ | Binary: $01000101$
  • r: Decimal $114$ | Binary: $01110010$
  • i: Decimal $105$ | Binary: $01101001$
  • c: Decimal $99$ | Binary: $01100011$
  • k: Decimal $107$ | Binary: $01101011$
  • S: Decimal $83$ | Binary: $01010011$
  • i: Decimal $105$ | Binary: $01101001$
  • n: Decimal $110$ | Binary: $01101110$
  • g: Decimal $103$ | Binary: $01100111$
  • h: Decimal $104$ | Binary: $01101000$

• Binary Mapping Grid (Reference):

  • Columns: $128, 64, 32, 16, 8, 4, 2, 1$
  • Example (E = 69): $0 \times 128 + 1 \times 64 + 0 \times 32 + 0 \times 16 + 0 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1 = 69$

Questions & Discussion

• Question from Student: "But couldn’t I get to 9 like this, Binary-Bot? [Using decimals 4 and 1 in the grid] Or like this? [1x4 and 2x2 and 1x1]?"
• Binary-Bot Response: "No! Binary only uses 1s and 0s."