Grade 9

Intermediate

8 min

Lesson 3 of 12

Encoding Numbers: Decimal to Binary

Not Started

Master the technique of converting standard numbers into binary format for computer processing.

Every photo you take and every text you send is secretly a massive string of 1s and 0s. But how does a computer translate a number like 150 into a language made only of two digits?

Learning Objectives

1.Convert decimal numbers between 0 and 255 into 8-bit binary strings
2.Apply the subtraction method to translate base-10 numbers into base-2
3.Determine the minimum number of bits required to represent any specific decimal value

The Binary Grid

To understand binary, we first need to look at place values. In our standard decimal system (Base-10), each position is a power of 10 (

10^0, 10^1, 10^2

). In Binary (Base-2), each position is a power of 2. For an 8-bit string (a byte), the values from right to left are:

128, 64, 32, 16, 8, 4, 2, 1

These correspond to

2^7

down to

2^0

. To represent a number, we simply decide which of these 'slots' are turned ON (1) or OFF (0). If the slot is ON, we add its value to our total.

Let's convert the decimal number 5 into binary. 1. Look at our grid: $128, 64, 32, 16, 8, 4, 2, 1$ . 2. 5 is smaller than 8, so the first five slots are 0. 3. We need a 4 ( $2^2$ ), so we put a 1 there. Remainder: $5 - 4 = 1$ . 4. We don't need a 2, so that's a 0. 5. We need a 1 ( $2^0$ ), so we put a 1 there. 6. Result: $00000101$ .

Quick Check

What is the decimal value of the binary position $2^4$ ?

Multiply 2 by itself four times:

2 \times 2 \times 2 \times 2

Answer

The Subtraction Method

The most reliable way to convert larger numbers is the Subtraction Method. You start with your decimal number and look at the 8-bit place values ( $128, 64, 32, 16, 8, 4, 2, 1$ ). Ask: 'Can I subtract this place value from my current total?' If yes, write a 1 and subtract the value. If no, write a 0 and move to the next smaller place value. You repeat this until you reach the $2^0$ position. This systematic approach ensures you never miss a bit or double-count a value.

Convert 172 to 8-bit binary: 1. Can we subtract 128 from 172? Yes. ( $172 - 128 = 44$ ). Bit: 1 2. Can we subtract 64 from 44? No. Bit: 0 3. Can we subtract 32 from 44? Yes. ( $44 - 32 = 12$ ). Bit: 1 4. Can we subtract 16 from 12? No. Bit: 0 5. Can we subtract 8 from 12? Yes. ( $12 - 8 = 4$ ). Bit: 1 6. Can we subtract 4 from 4? Yes. ( $4 - 4 = 0$ ). Bit: 1 7. Remaining bits (2 and 1) are 0. Result: $10101100$

Quick Check

If a decimal number is odd (like 7, 15, or 21), what will the very last bit (the $2^0$ place) always be?

Think about whether any other power of 2 can create an odd total.

Answer

Bit Capacity and Limits

How many numbers can we actually fit into 8 bits? The largest value occurs when all bits are 1 ( $11111111$ ). This equals $128+64+32+16+8+4+2+1 = 255$ . A helpful rule is that $n$ bits can represent $2^n$ unique values (including zero), ranging from $0$ to $2^n - 1$ . If you need to represent the number 300, 8 bits won't be enough because $255 < 300$ . You would need to add a 9th bit ( $2^8 = 256$ ) to handle it.

How many bits are needed to represent the decimal number 40? 1. List powers of 2: $1, 2, 4, 8, 16, 32, 64$ . 2. 32 is the largest power less than 40. This is the 6th position ( $2^5$ ). 3. Therefore, we need at least 6 bits to represent 40. 4. Calculation: $32 + 8 = 40$ , which is $101000$ in binary.

Key Takeaways

1Binary place values follow powers of two: $128, 64, 32, 16, 8, 4, 2, 1$ .
2The Subtraction Method involves placing a '1' if a place value fits and a '0' if it doesn't.
3An 8-bit byte can represent any integer from 0 to 255.
4To find the bits needed for a number $x$ , find the smallest $n$ such that $2^n > x$ .

Quick Check

Multiple Choice

What is the 8-bit binary representation of the decimal number 10?

Multiple Choice

What is the maximum decimal value that can be represented by 4 bits?

True/False

The binary string 11000000 is equal to the decimal number 192.

What's Next?

Review Tomorrow

In 24 hours, try to write down the eight place values for an 8-bit number from memory and convert your current age into binary.

Practice Activity

Find three small items near you (like pens or coins). Assign them values of 4, 2, and 1. Try to 'show' the numbers 1 through 7 by picking up the items.

Browse

Browse

Encoding Numbers: Decimal to Binary

Learning Objectives

The Binary Grid

The Subtraction Method

Bit Capacity and Limits

Key Takeaways

Quick Check

What's Next?

Encoding Numbers: Decimal to Binary

Learning Objectives

The Binary Grid

The Subtraction Method

Bit Capacity and Limits

Key Takeaways

Quick Check

What's Next?