← ACARA Digital Technologies Years 7–8

Converting decimal to binary by repeated division by 2

Divide a number by 2 again and again until the quotient is 0, noting each remainder. Read the remainders from last to first to get the binary number. Example: 13 is 1101.

Part of: ACARA Digital Technologies Years 7–8 — Binary Number Representation — Module 21

Understanding Converting decimal to binary by repeated division by 2 in ACARA Digital Technologies Years 7–8

What it does: Reading the remainders 1, 0, 1, 1 backwards (bottom to top) gives 1101, which is 13 in binary. 13 ÷ 2 = 6 r1, 6 ÷ 2 = 3 r0, 3 ÷ 2 = 1 r1, 1 ÷ 2 = 0 r1. The remainders come out as 1, 0, 1, 1 and are then read in reverse. The last remainder is the most significant bit, so you read the remainders from the bottom up. Keep dividing until the quotient is 0. The final division, 1 ÷ 2, gives a quotient of 0 and a remainder of 1. Dividing by 2 leaves a remainder of 0 (even) or 1 (odd). That is why each remainder is a single binary digit.

Common mistakes: 6 is even, so it divides by 2 exactly and the remainder is 0, not 1. The line should be 6 / 2 = 3 remainder 0. 5 divided by 2 is 2 with a remainder of 1, not 3. The line should be 5 / 2 = 2 remainder 1.

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This module's key terms are also in a printable crossword (Modules 189–203) and word search (Modules 189–236).

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Learning
Duo
When converting 13 to binary by repeated division, which remainders do you get, in the order you calculate them?
1, 1, 0, 11, 0, 1, 1
Matches
13 → remainders 1, 0, 1, 1 → answer 1011
↓
The remainders were read from first to last (top to bottom) instead of last to first. The correct answer is 1101.
Applying
Meanings
Term
repeated division by 2
Meaning
Divide a number by 2 again and again until the quotient is 0, noting each remainder. Read the remainders from last to first to get the binary number. Example: 13 is 1101.
Practice
number ← 13
Testing
Gaps
13 / = 6 remainder 1
Check Answer
Spots
113 / 2 = 6 remainder 1
26 / 2 = 3 remainder 1

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Quick Questions

1. When converting 13 to binary by repeated division, which remainders do you get, in the order you calculate them?
A. 1, 1, 0, 1
B. 1, 0, 1, 1 ✓
13 ÷ 2 = 6 r1, 6 ÷ 2 = 3 r0, 3 ÷ 2 = 1 r1, 1 ÷ 2 = 0 r1. The remainders come out as 1, 0, 1, 1 and are then read in reverse.
2. How do you read the remainders to write the binary number?
A. From last to first (bottom to top) ✓
B. From first to last (top to bottom)
The last remainder is the most significant bit, so you read the remainders from the bottom up.
3. When do you stop dividing by 2?
A. When the quotient reaches 0 ✓
B. When the remainder reaches 0
Keep dividing until the quotient is 0. The final division, 1 ÷ 2, gives a quotient of 0 and a remainder of 1.
4. Which remainders are possible when you divide by 2?
A. Any digit from 0 to 9
B. Only 0 or 1 ✓
Dividing by 2 leaves a remainder of 0 (even) or 1 (odd). That is why each remainder is a single binary digit.
5. What is 6 in binary using repeated division by 2?
A. 110 ✓
B. 011
6 ÷ 2 = 3 r0, 3 ÷ 2 = 1 r1, 1 ÷ 2 = 0 r1. Reading the remainders from last to first gives 110.

Match the Pairs

13 ÷ 2 = 6 remainder 1 → The first division step for 13: the quotient is 6 and the remainder 1 is noted. This remainder becomes the last binary digit (the least significant bit).
1 ÷ 2 = 0 remainder 1 → The final division step: the quotient reaches 0, so you stop. This last remainder is the first binary digit (the most significant bit).
Read the remainders from last to first (bottom to top) to build the binary number. For 13 the remainders in order are 1, 0, 1, 1. → 1101
The place-value method gives the same answer as repeated division. 13 = 8 + 4 + 0 + 1, so use the places 8, 4, 2, 1. → 8 + 4 + 1 = 13 → 1101
13 → remainders 1, 0, 1, 1 → answer 1011 → The remainders were read from first to last (top to bottom) instead of last to first. The correct answer is 1101.
13 ÷ 2 = 6, 6 ÷ 2 = 3, 3 ÷ 2 = 1 → stop → The division stopped when the quotient was 1. You must keep dividing until the quotient is 0 (1 ÷ 2 = 0 remainder 1), or the leftmost bit is missing.

Key Term

repeated division by 2
Divide a number by 2 again and again until the quotient is 0, noting each remainder. Read the remainders from last to first to get the binary number. Example: 13 is 1101.

Fill in the Blanks

1. 13 / 2 = 6 remainder 1
Hint: Type the number you keep dividing by when converting decimal to binary.
2. 13 / 2 = 6 remainder 1 6 / 2 = 3 remainder 0
Hint: Type the remainder when 6 is divided by 2.
3. 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1 quotient is 0, so stop
Hint: Type the quotient that tells you to stop dividing.
4. Read the remainders from last to first
Hint: Type the word for where you finish reading the remainders.
5. 8 + 4 + 1 = 13 so 13 is 1101
Hint: Type the binary number you get for 13, which matches the division method.

Spot the Error

Question 1
A 13 / 2 = 6 remainder 1
B 6 / 2 = 3 remainder 1
C 3 / 2 = 1 remainder 1
D 1 / 2 = 0 remainder 1
E Read bottom to top: 1101
Line B — 6 is even, so it divides by 2 exactly and the remainder is 0, not 1. The line should be 6 / 2 = 3 remainder 0.
Question 2
A 10 / 2 = 5 remainder 0
B 5 / 2 = 3 remainder 1
C 2 / 2 = 1 remainder 0
D 1 / 2 = 0 remainder 1
E Read bottom to top: 1010
Line B — 5 divided by 2 is 2 with a remainder of 1, not 3. The line should be 5 / 2 = 2 remainder 1.
Question 3
A 6 / 2 = 3 remainder 2
B 3 / 2 = 1 remainder 1
C 1 / 2 = 0 remainder 1
D Read bottom to top: 110
Line A — When you divide by 2 the remainder can only be 0 or 1. Since 6 divides exactly, the line should be 6 / 2 = 3 remainder 0.
Question 4
A 9 / 2 = 4 remainder 1
B 4 / 2 = 2 remainder 0
C 2 / 2 = 1 remainder 0
D 1 / 2 = 0 remainder 0
E Read bottom to top: 1001
Line D — 1 divided by 2 is 0 with a remainder of 1, because 2 goes into 1 zero times. The line should be 1 / 2 = 0 remainder 1.
Question 5
A n ← 11
B WHILE n > 1
C r ← n MOD 2
D n ← n DIV 2
E END WHILE
Line B — You must keep dividing until the quotient is 0. Stopping when n is 1 would miss the last remainder. It should be WHILE n > 0.

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