Binary number system in Intro to Computing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the binary number system?

The binary number system is a way of counting using only two digits: 0 and 1. It is the base-2 numeral system used by computers to represent data.

beginner

Why do computers use the binary number system?

Computers use binary because their electronic circuits have two states: ON and OFF. These two states are naturally represented by 1 and 0, making binary simple and reliable for computers.

intermediate

Convert the binary number 1011 to decimal.

1011 in binary equals 11 in decimal. Calculation: (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11.

beginner

What does each position in a binary number represent?

Each position in a binary number represents a power of 2, starting from 2⁰ at the rightmost digit, then 2¹, 2², and so on moving left.

intermediate

How do you add two binary numbers?

Add binary digits like decimal but with base 2: 0+0=0, 1+0=1, 1+1=10 (write 0 carry 1). Carry over when sum is 2 (10 in binary).

What digits are used in the binary number system?

A 1 to 10

B 0 to 9

C 0 and 1

D A and B

Binary uses only two digits: 0 and 1.

What is the decimal value of binary 110?

A 6

B 8

C 4

D 3

110 binary = (1×2²)+(1×2¹)+(0×2⁰) = 4 + 2 + 0 = 6.

Why is binary suitable for computers?

A Because it uses many digits

B Because it uses decimal numbers

C Because it is easier to read

D Because computers have two states: ON and OFF

Computers use binary because their circuits have two states: ON (1) and OFF (0).

What does the rightmost digit in a binary number represent?

A 2² (four's place)

B 2⁰ (one's place)

C 2¹ (two's place)

D 2³ (eight's place)

The rightmost digit represents 2⁰, which equals 1.

What is the result of binary addition 1 + 1?

A 0 with carry 1

B 2

C 1

D 10 with carry 0

1 + 1 in binary equals 10, so write 0 and carry 1 to the next digit.

Explain how to convert a binary number to decimal with an example.

Think about how each binary digit represents a power of 2 and add them up.

Describe why computers use the binary number system instead of decimal.

Consider the physical nature of computer hardware.

Practice

1. What digits are used in the binary number system?

easy

A. Digits 1 to 10

B. Digits 0 to 9

C. Digits 0 to 7

D. Only 0 and 1

Solution

1. Step 1: Understand the binary system basics

   The binary number system uses only two digits to represent all numbers.

2. Step 2: Identify the digits used

   These digits are 0 and 1, representing off and on states in computers.

3. Final Answer:

   Only 0 and 1 -> Option D

4. Quick Check:

   Binary digits = 0 and 1 [OK]

Hint: Binary uses just two digits: 0 and 1 [OK]

Common Mistakes:

• Confusing binary with decimal digits
• Thinking binary uses digits 0 to 9
• Assuming binary includes digits beyond 1

2. Which of the following is the correct binary number?

easy

A. 1101

B. 1234

C. 1021

D. 2010

Solution

1. Step 1: Check each digit for binary validity

   Binary numbers only contain 0s and 1s. Check each option's digits.

2. Step 2: Identify the valid binary number

   1101 is 1101, which contains only 1s and 0s. Others have digits like 2 or 3, invalid in binary.

3. Final Answer:

   1101 -> Option A

4. Quick Check:

   Valid binary = digits 0 or 1 only [OK]

Hint: Binary digits are only 0 or 1, no other digits allowed [OK]

Common Mistakes:

• Choosing numbers with digits other than 0 or 1
• Confusing binary with decimal or octal numbers
• Ignoring invalid digits in options

3. What is the decimal value of the binary number 1011?

medium

A. 7

B. 11

C. 13

D. 9

Solution

1. Step 1: Assign powers of 2 to each bit

   From right to left, bits represent 2^0=1, 2^1=2, 2^2=4, 2^3=8.

2. Step 2: Calculate decimal value by adding bits with 1

   Bits: 1(8) + 0(4) + 1(2) + 1(1) = 8 + 0 + 2 + 1 = 11.

3. Final Answer:

   11 -> Option B

4. Quick Check:

   Binary 1011 = Decimal 11 [OK]

Hint: Add powers of 2 where bit is 1, starting from right [OK]

Common Mistakes:

• Misaligning bit positions and powers of 2
• Adding all bits instead of weighted values
• Confusing binary digits with decimal digits

4. Identify the error in this binary number: 11012

medium

A. Binary numbers cannot start with 1

B. Too many digits for a binary number

C. Contains digit '2' which is invalid in binary

D. Missing leading zeros

Solution

1. Step 1: Check each digit for binary validity

   Binary digits must be only 0 or 1. The digit '2' is not allowed.

2. Step 2: Confirm the invalid digit

   The presence of '2' makes the number invalid as a binary number.

3. Final Answer:

   Contains digit '2' which is invalid in binary -> Option C

4. Quick Check:

   Binary digits = 0 or 1 only [OK]

Hint: Binary digits never include 2 or higher [OK]

Common Mistakes:

• Thinking binary numbers have digit limits on length
• Believing binary cannot start with 1
• Assuming missing zeros is an error

5. Convert the decimal number 18 to binary.

hard

A. 10010

B. 11001

C. 10110

D. 10001

Solution

1. Step 1: Divide decimal number by 2 repeatedly

   18 ÷ 2 = 9 remainder 0; 9 ÷ 2 = 4 remainder 1; 4 ÷ 2 = 2 remainder 0; 2 ÷ 2 = 1 remainder 0; 1 ÷ 2 = 0 remainder 1.

2. Step 2: Write remainders in reverse order

   Reading remainders from last to first: 1 0 0 1 0.

3. Final Answer:

   10010 -> Option A

4. Quick Check:

   Decimal 18 = Binary 10010 [OK]

Hint: Divide by 2, collect remainders backward [OK]

Common Mistakes:

• Reading remainders in wrong order
• Mixing up division and subtraction steps
• Confusing binary digits with decimal digits