Convert between positive denary and positive binary

🔢 Data Representation: Converting Between Positive Denary and Positive Binary

What is Denary (Decimal)?

Denary is the number system we use every day. It has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Think of it as a “10‑base” system because each place value is a power of 10. For example, the number 345 means: $$3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0.$$

What is Binary?

Binary uses only two digits: 0 and 1. It’s a “2‑base” system, so each place value is a power of 2. For instance, the binary number 1011 means: $$1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0.$$

Why Convert?

Computers store everything in binary, but we humans read decimal. Converting lets us understand how a computer represents the numbers we use every day. It’s like translating a message from one language to another.

Converting Denary ➜ Binary

1. Take the denary number you want to convert.
2. Divide it by 2. Write down the remainder (0 or 1).
3. Replace the number with the quotient and repeat step 2 until the quotient is 0.
4. Read the remainders from bottom to top – that’s the binary number.

Example: Convert 13 to binary.

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

Converting Binary ➜ Denary

1. Write the binary digits with their place values (powers of 2).
2. Multiply each digit by its power of 2.
3. Sum all the products to get the denary value.

Example: Convert 1101 to denary.

• $1 \times 2^3 = 8$
• $1 \times 2^2 = 4$
• $0 \times 2^1 = 0$
• $1 \times 2^0 = 1$
• Sum: $8 + 4 + 0 + 1 = 13$

Practice Problems

1. Convert 27 to binary.
2. Convert 1010 to denary.
3. Convert 45 to binary.
4. Convert 111100 to denary.

Answers