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Number Systems: Binary and Decimal Conversions

SS1 Term 1 • Week 2
About 45 minutes
Learning goals: Explain positional notation across Base 10 (Decimal) and Base 2 (Binary). • Convert positive integers from Decimal to Binary using the Repeated Division by 2 method. • Convert Binary numbers to Decimal using the Positional Weight expansion method. • Calculate powers of 2 ($2^0$ to $2^{10}$) and explain bit, nibble, byte, and word units.

Why do humans count in Base 10 (Decimal)? The answer is sitting at the end of your arms: humans have ten fingers! If human beings had been born with eight fingers instead of ten, our entire global mathematical civilization would count in Base 8!

Computers, on the other hand, do not have fingers. They have transistors that can only be Charged (1) or Discharged (0). Therefore, all digital computers count, calculate, store photographs, and play music using Base 2 (Binary).

In this lesson, we will master the mathematical mechanics of converting numbers back and forth between Decimal (Base 10) and Binary (Base 2).

Learning Outcomes

By the end of this lesson, you will be able to:

  1. Explain positional number notation using powers of the base ($b^0, b^1, b^2...$).
  2. Convert any positive decimal integer to binary using Repeated Division by 2.
  3. Convert any binary sequence to decimal using Positional Weight Expansion.
  4. Identify standard data units: Bit, Nibble (4 bits), Byte (8 bits), and Word.

Key Vocabulary

Term Mathematical Meaning
Radix (Base) The number of unique digits used in a positional number system (Base 10 uses 0–9; Base 2 uses 0–1).
Bit Binary Digit — the smallest indivisible unit of data in computing (either 0 or 1).
Nibble A group of 4 binary bits (e.g., 1010).
Byte A group of 8 binary bits (e.g., 11001010).
Most Significant Bit (MSB) The leftmost bit in a binary number, carrying the highest positional value.
Least Significant Bit (LSB) The rightmost bit in a binary number, carrying the lowest positional value ($2^0 = 1$).

Powers of 2 Reference Table

Memorizing the powers of 2 from $2^0$ to $2^8$ will make you lightning-fast at computing mathematics:

Power $2^7$ $2^6$ $2^5$ $2^4$ $2^3$ $2^2$ $2^1$ $2^0$
Value 128 64 32 16 8 4 2 1

Method 1: Binary to Decimal (Positional Expansion)

To convert a binary number to decimal, multiply each binary digit by its corresponding power of 2, then sum the products:

Example: Convert $(110101)_2$ to Decimal

$$\begin{aligned} (110101)2 &= (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) \ &= (1 \times 32) + (1 \times 16) + (0 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) \ &= 32 + 16 + 0 + 4 + 0 + 1 \ &= \mathbf{53 \end{aligned}$$}

Method 2: Decimal to Binary (Repeated Division by 2)

To convert a decimal number to binary: 1. Divide the decimal number by 2. 2. Record the integer quotient and the remainder (which will always be either 0 or 1). 3. Repeat dividing the quotient until the quotient reaches 0. 4. The Golden Rule: Read the remainders from bottom to top (from MSB to LSB)!

Example: Convert $43_{10}$ to Binary

Division Step       Quotient    Remainder
-----------------------------------------
43 / 2      =          21       Remainder 1  ^ (LSB)
21 / 2      =          10       Remainder 1  |
10 / 2      =           5       Remainder 0  |  READ
 5 / 2      =           2       Remainder 1  |  UPWARDS
 2 / 2      =           1       Remainder 0  |  (Bottom to Top!)
 1 / 2      =           0       Remainder 1  | (MSB)

Reading remainders from bottom to top:

$$43_{10} = \mathbf{101011_2}$$

Verification:

$$32 + 0 + 8 + 0 + 2 + 1 = 43_{10} \quad \checkmark$$

Practical Class Activity: Binary Arithmetic Drills

Solve the following four problems in your notebook:

  1. Convert to Decimal:
  2. a) $(1111)2 = 8 + 4 + 2 + 1 = \mathbf{15$}
  3. b) $(10000000)2 = \mathbf{128$}
  4. Convert to Binary:
  5. a) $18_{10} = 16 + 2 = \mathbf{10010_2}$
  6. b) $60_{10} = 32 + 16 + 8 + 4 = \mathbf{111100_2}$

Common Mistakes & Corrections

Common Mistake Why It Is Incorrect Proper Correction
"Reading division remainders from top to bottom." The first remainder produced represents the units column ($2^0$ - LSB). Always read remainders in reverse: from the bottom (MSB) up to the top (LSB).
"Assuming $2^0 = 0$." Any non-zero number raised to the power of zero is mathematically equal to 1. Remember: $2^0 = 1$.

Notebook Summary

Copy Into Your Computing Notebook

  • Binary (Base 2): Uses digits 0 and 1, with column weights: $1, 2, 4, 8, 16, 32, 64, 128...$
  • Binary to Decimal: Multiply each bit by its positional power of 2 and sum the results.
  • Decimal to Binary: Repeatedly divide by 2 and read remainders bottom-to-top.
  • 1 Nibble = 4 bits; 1 Byte = 8 bits.

Check your understanding

What is the decimal (Base 10) value of the binary number (1011)₂?
What is the binary representation of the decimal number 25?
When converting a decimal number to binary using repeated division by 2, how should the recorded remainders be read to form the final binary number?
How many bits are contained in a 'Nibble' and a 'Byte' respectively?
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