Converting Between Decimal & Binary

• Decimal (Base 10): The standard numbering system used by humans, consisting of ten digits ($0$ through $9$). Each position in a decimal number represents a power of $10$, such as units ($10^0$), tens ($10^1$), and hundreds ($10^2$).

• Binary (Base 2): The fundamental language of computers, consisting of only two digits: $0$ and $1$. Each position represents a power of $2$, such as $1, 2, 4, 8, 16, 32, 64,$ and $128$.

• Bit and Byte: A single binary digit is called a bit. A group of eight bits is known as a byte, which is the standard unit for measuring data capacity in computing.

Binary to Decimal Conversion

• Step 1: Write out the binary place values (headings) from right to left, starting at $1$ and doubling each time ($1, 2, 4, 8, \dots$).

• Step 2: Place the binary digits under their corresponding headings.

• Step 3: Sum the values of all headings that have a '1' beneath them to find the total decimal value.

Decimal to Binary Conversion

• Step 1: List the binary headings that are less than or equal to the decimal number.

• Step 2: Starting from the largest heading, determine if it 'fits' into the decimal number. If it does, place a '1' and subtract that heading's value from the decimal total.

• Step 3: If it does not fit, place a '0' and move to the next smaller heading. Repeat until all headings are filled.

• The Odd/Even Check: Always check the rightmost bit (the $1$s column). If the binary number ends in $1$, the decimal equivalent must be an odd number; if it ends in $0$, it must be even.

• Show Your Workings: In exams, marks are often awarded for the process. Even if the final answer is wrong, showing the subtraction steps or the addition of place values can secure partial credit.

• Leading Zeros: Be careful with leading zeros in 8-bit representations. Ensure you fill all 8 positions even if the number is small (e.g., $5$ should be written as $00000101$ if an 8-bit byte is required).

• Sanity Check: If you are converting a number like $130$ and your binary result has a $0$ in the $128$ column, you know immediately that the answer is incorrect.

• Reversing Bit Order: A common mistake is writing the binary number in reverse (Least Significant Bit on the left). Always ensure the $128$ column is on the far left and the $1$ column is on the far right.

• Calculation Errors: Simple addition or subtraction errors are the most frequent cause of lost marks. Double-check your subtractions during the decimal-to-binary process.

• Off-by-One Errors: Students often forget that $2^0 = 1$. Starting the place values at $2$ instead of $1$ will shift every subsequent value and result in an incorrect conversion.