Lesson 6: Number System Conversion

Types of Number System Conversions

• Binary to Decimal, Octal, Hexadecimal

• Decimal to Binary, Octal, Hexadecimal

• Octal to Decimal, Binary, Hexadecimal

• Hexadecimal to Binary, Octal, Decimal

2

Binary numbers are based on the radix of __.

10

Decimal numbers are based on the radix of ___.

1, 0

Binary will only be represented in__ and ___

2

What is the base of binary numbers?

10

What is the base of decimal numbers?

1 and 0

What digits does binary use?

Write exponent markers above the digits starting from 0 on the right.

What is the first step in converting binary to decimal?

The power of 2 for each digit.

What does the exponent marker show?

Multiply them by 2 raised to the power of their position.

What do you do with the digits that are 1?

Ignore them (they contribute 0).

What do you do with the digits that are 0?

Add all the results together.

What is the last step after multiplying the digits?

binary is based on 2.

Why do we use powers of 2?

13₁₀ (8 + 4 + 0 + 1)

What is the decimal value of 1101₂

Binary to Octal Conversion

It is converting binary numbers to octal

3 binary digits

How many binary digits make up one octal digit?

Group the binary digits into groups of three, starting from the right (least significant bit).

What is the first step when converting binary to octal?

Add 0s to the left until it has three digits.

What do you do if the leftmost group doesn’t have three digits?

Add 0s to the right to complete a group of three digits.

What should you do if the number has a decimal part (fraction)?

Convert each group of three binary digits into its decimal equivalent.

After grouping into threes, what’s the next step?

What is the place value concept used when converting each group?

(4, 2, 1) – each position represents 4, 2, and 1 from left to right.

What do you do after converting each group into decimal?

Write the decimal results together — that’s the octal number.

Because one octal digit equals three binary digits.

Why do we group binary digits into three?

Binary to Hexadecimal Conversion

It is the convertion of binary values to hexadecimal conversion

4 binary digits

How many binary digits make up one hexadecimal digit?

Group the binary digits into groups of four, starting from the right (least significant bit).

What is the first step when converting binary to hexadecimal?

Add 0s to the left until it has four digits.

What do you do if the leftmost group doesn’t have four digits?

Add 0s to the right until the group has four digits.

What should you do if the binary number has a fractional part (decimal)?

Convert each group of four binary digits into its decimal equivalent.

After grouping into fours, what’s the next step?

(8, 4, 2, 1) – each position from left to right represents 8, 4, 2, and 1.

What place value concept is used when converting each group of four?

Replace them with their hexadecimal symbols: A=10, B=11, C=12, D=13, E=14, F=15.

What should you do with decimal results from 10 to 15?

Combine them to form the final hexadecimal number.

What do you do after converting all groups into hex digits?

• Step 1: Group → 1101 0110

• Step 2: Convert → 13 6

• Step 3: Replace → D6₁₆

Example: What is 11010110₂ in hexadecimal?

Because one hexadecimal digit equals four binary digits.

Why do we group binary digits into four?

Decimal to Binary Conversion

It is the convertion of decimal numbers to binary values.

To change a number with base 10 into a number with base 2.

What is the goal of decimal to binary conversion?

Successive division by 2.

What is the main method used to convert the whole number part?

Successive multiplication by 2.

What is the main method used to convert the fractional part (if any)?

Divide the decimal number by 2.

What is the first step when converting a decimal number to binary?

Record the remainder — it will become part of the binary number.

What do you do with the remainder after dividing by 2?

Divide the quotient again by 2.

What do you do with the quotient after division?

When the quotient becomes 0.

When do you stop dividing by 2?

Write the remainders in reverse order (from last to first).

How do you write the final binary number?

Multiply the fractional part by 2 repeatedly and record the whole number parts that appear.

What should you do if there’s a fractional part in the decimal number?

Combine the whole number result (from division) and the fractional result (from multiplication) separated by a binary point.

How do you form the final binary number when there’s a fraction?

25 ÷ 2 = 12 remainder 1

12 ÷ 2 = 6 remainder 0

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Final answer (reverse remainders): 11001₂

Convert 25₁₀ to binary

Because the first remainder corresponds to the least significant bit (LSB) and the last remainder is the most significant bit (MSB).

Why do we reverse the order of the remainders?

Decimal to Octal Conversion

It is the conversion from decimal to octal values.

To convert a number from base 10 (decimal) to base 8 (octal).

What is the goal of decimal to octal conversion?

Successive division by 8.

What is the main method used to convert the whole number part?

Successive multiplication by 8.

What is the method used to convert the fractional part (if any)?

Divide the decimal number by 8.

What is the first step in converting a decimal number to octal?

Record the remainder — it becomes part of the octal result.

What do you do with the remainder after division?

What do you do with the quotient?

Continue dividing the quotient by 8.

When the quotient becomes 0.

When do you stop dividing by 8?

How do you write the final octal number?

Write the remainders in reverse order (from last to first).

Multiply the fractional part by 8 and record the whole number part each time.

How do you convert a fractional part to octal?

Combine the whole number result (from division) and the fractional result (from multiplication) with a radix point (octal point).

How do you form the final octal result if there’s a fraction?

• 125 ÷ 8 = 15 remainder 5

• 15 ÷ 8 = 1 remainder 7

• 1 ÷ 8 = 0 remainder 1

• Final answer (reverse order): 175₈

Convert 125₁₀ to octal.

Because the first remainder represents the least significant digit (rightmost) and the last remainder is the most significant (leftmost).

Why is the remainder reversed in the final answer?

To convert a number from base 10 (decimal) to base 16 (hexadecimal).

What is the goal of decimal to hexadecimal conversion?

Successive division by 16.

What is the main method used to convert the whole number part?

Successive multiplication by 16.

What is the method used to convert the fractional part (if any)?

Divide the decimal number by 16.

What is the first step in converting a decimal number to hexadecimal?

Record the remainder — it becomes part of the hexadecimal result.

What do you do with the remainder after dividing by 16?

Continue dividing the quotient by 16.

What do you do with the quotient after each division?

When the quotient becomes 0.

When do you stop dividing by 16?

Write the remainders in reverse order (from last to first).

How do you write the final hexadecimal number?

Replace it with its hexadecimal symbol:

• 10 → A

• 11 → B

• 12 → C

• 13 → D

• 14 → E

• 15 → F

What do you do if a remainder is between 10 and 15?

Multiply the fractional part by 16 and record the whole number part each time.

How do you convert a fractional part to hexadecimal?

Combine the whole number result (from division) and the fractional result (from multiplication) with a hexadecimal point.

How do you form the final hexadecimal number if there’s a fraction?

• 250 ÷ 16 = 15 remainder 10 (A)

• 15 ÷ 16 = 0 remainder 15 (F)

• Final answer (reverse order): FA₁₆

Example: Convert 250₁₀ to hexadecimal.

Why do we reverse the order of remainders in the final answer?

Because the first remainder represents the least significant digit, while the last remainder is the most significant.