← ACARA Digital Technologies Years 7–8

Common mistakes when converting between decimal and binary

A slip such as reversing place values, dropping a zero, reading remainders top to bottom or using a 2. Binary only uses 0 and 1, and remainders are read bottom to top.

Part of: ACARA Digital Technologies Years 7–8 — Binary Number Representation — Module 23

Understanding Common mistakes when converting between decimal and binary in ACARA Digital Technologies Years 7–8

What it does: The place values 8 4 2 1 line up with the bits from left to right, so the 1 sits at the right-hand (least significant) end. The last remainder is the most significant bit. Reading 1, 1, 0 from the bottom up gives 110, which is 6 in binary. 9 = 8 + 1, so the 4 and 2 columns need a 0. Dropping the middle zero gives 101, which is 5. A 4-bit number needs all four bits, so the leading zero must stay. 101 is only three bits. Binary only uses the digits 0 and 1. A 2 is never a binary digit.

Common mistakes: The place values were written from the wrong end. In binary the leftmost bit has the biggest value, so they should be 8 4 2 1. A binary digit can only be 0 or 1. The 2s place needs one 2, so it holds the digit 1, not the digit 2.

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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
A student converts 6 to 3-bit binary. Which answer is correct?
110011
Matches
110 = 1x1 + 1x2 + 0x4 = 3
↓
The place values started from the left end. The 1 place belongs on the right, so 110 = 1x4 + 1x2 + 0x1 = 6.
Applying
Meanings
Term
binary conversion mistake
Meaning
A slip such as reversing place values, dropping a zero, reading remainders top to bottom or using a 2. Binary only uses 0 and 1, and remainders are read bottom to top.
Practice
13 / 2 = 6 remainder 1
Testing
Gaps
Place values right to left: 1, 2, 4,
Check Answer
Spots
1Binary 1101 to decimal
2Place values: 1 2 4 8

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Quick Questions

1. A student converts 6 to 3-bit binary. Which answer is correct?
A. 110 ✓
B. 011
6 = 4 + 2, so the 4 and 2 columns get a 1. 011 is 3, because the place values were started from the wrong end.
2. Which list shows binary place values going from the right-hand bit to the left?
A. 1, 2, 4, 8 ✓
B. 8, 4, 2, 1
The place values start at 1 on the right and double each step to the left. Starting from the wrong end is a common mistake.
3. Write 9 in binary. Which answer is correct?
A. 1001 ✓
B. 101
9 = 8 + 1, so the 4 and 2 columns need a 0. Dropping the middle zero gives 101, which is 5.
4. Write 5 as a 4-bit binary number. Which answer is correct?
A. 101
B. 0101 ✓
A 4-bit number needs all four bits, so the leading zero must stay. 101 is only three bits.
5. Dividing 13 by 2 again and again gives remainders 1, 0, 1, 1 in the order you find them. How do you read them?
A. Top to bottom: 1011
B. Bottom to top: 1101 ✓
The last remainder is the most significant bit, so read the remainders from bottom to top. Reading downwards gives 1011, which is 11.

Match the Pairs

1001 written as 101 → A middle zero was dropped, so 9 became 5
1201 → Invalid binary: 2 is not a binary digit
Binary place values start at 1 on the right and double as you move left → 8 4 2 1
When dividing by 2, read the remainders from bottom to top to get the binary number → 6/2=3 r0, 3/2=1 r1, 1/2=0 r1 -> 110
110 = 1x1 + 1x2 + 0x4 = 3 → The place values started from the left end. The 1 place belongs on the right, so 110 = 1x4 + 1x2 + 0x1 = 6.
5 in 8-bit binary: 101 → The leading zeros are missing. In 8 bits, 5 should be written 00000101.

Key Term

binary conversion mistake
A slip such as reversing place values, dropping a zero, reading remainders top to bottom or using a 2. Binary only uses 0 and 1, and remainders are read bottom to top.

Fill in the Blanks

1. Place values right to left: 1, 2, 4, 8
Hint: Type the next binary place value. Each one doubles, and you start from the right-hand end.
2. Read remainders from bottom to top
Hint: Type the word for where you start reading the remainders after repeated division by 2.
3. 9 = 8 + 1, so 4-bit binary is 1001
Hint: Type the digit for the middle place values that are not used, so no zero gets dropped.
4. Binary 1021 is wrong because of the digit 2
Hint: Type the digit that can never appear in a binary number.
5. Remainders down: 1, 0, 1, 1 Read upwards: 1101
Hint: Type the binary number you get by reading the remainders from bottom to top.

Spot the Error

Question 1
A Binary 1101 to decimal
B Place values: 1 2 4 8
C 8 + 4 + 0 + 1 = 13
D Answer: 13
Line B — The place values were written from the wrong end. In binary the leftmost bit has the biggest value, so they should be 8 4 2 1.
Question 2
A Convert 6 to 3-bit binary
B 6 = 4 + 2 so 4s=1, 2s=2
C 1s place = 0
D Answer: 110
Line B — A binary digit can only be 0 or 1. The 2s place needs one 2, so it holds the digit 1, not the digit 2.
Question 3
A Convert 6 to binary
B 6 / 2 = 3 remainder 0
C 3 / 2 = 1 remainder 1
D Read remainders top to bottom
E Answer: 110
Line D — Remainders must be read from bottom to top. Reading them top to bottom gives 011, which is the wrong order. It should be 110.
Question 4
A 9 = 8 + 1 = 101 in binary
B As 8 bits: 00001001
C Four zeros pad it to 8 bits
Line A — A middle zero was dropped. The 4s and 2s places are both 0, so 9 is 1001, not 101.
Question 5
A 10 / 2 = 5 remainder 0
B 5 / 2 = 2 remainder 1
C 2 / 2 = 1 remainder 0
D 1 / 2 = 0 remainder 2
E Bottom to top: 1010
Line D — When you divide by 2 the remainder can only be 0 or 1. 1 / 2 gives 0 remainder 1, and that 1 is the top bit of 1010.

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