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Number Systems, Binary Arithmetic and Data Representation Short Notes for GATE CS

Number Systems, Binary Arithmetic and Data Representation short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus s

number systems binary arithmetic and data representation short notes

Summary: The Radix Conversion Checklist

.smart-card { font-family: system-ui, sans-serif; padding: 1.5rem; background: var(--color-bg); color: var(--color-text); border-radius: 12px; border: 2px solid var(--color-primary); } .checklist { list-style: none; padding: 0; margin: 1rem 0; } .checklist li { margin-bottom: 0.75rem; display: flex; align-items: flex-start; gap: 0.75rem; font-size: 0.95rem; } .check-icon { color: var(--color-primary); font-weight: bold; } .highlight { color: var(--color-primary); font-weight: 600; } Revision Checklist 1. Validity Check: Are all digits < r ? 2. Identify Bases: Source and Target? 3. Power-of-2? Use bit grouping (3 or 4 bits). 4. Otherwise: Route through Base 10 . 5. Unknown r : Solve polynomial, verify r > max (d_{i}) .

Summary: Fractional Conversion Checklist

.sc-summary { font-family: system-ui, sans-serif; padding: 1.5rem; background: var(--color-bg); color: var(--color-text); border-radius: 12px; border: 2px solid var(--color-primary); } .sc-summary h3 { margin-top: 0; font-size: 1.15rem; color: var(--color-primary); } .sc-checklist { list-style: none; padding: 0; margin: 1rem 0 0; } .sc-checklist li { display: flex; align-items: flex-start; gap: 0.75rem; margin-bottom: 0.75rem; font-size: 0.95rem; line-height: 1.4; } .sc-check-icon { flex-shrink: 0; width: 20px; height: 20px; background: var(--color-primary); color: var(--color-bg); border-radius: 4px; display: flex; align-items: center; justify-content: center; font-size: 0.75rem; font-weight: bold; margin-top: 2px; } .sc-checklist strong { color: var(--color-text); } Final Revision Checklist 1 Radix Point: Integer weights are positive powers (right to left); fractional weights are negative powers (left to right). 2 Binary → Decimal: Sum . Rely on memorized values: . 3 Decimal → Binary: Repeated multiply by 2. Read integer parts top-down. 4 Non-Terminating: If it doesn't terminate, check for bit-limits or closest approximations. 5 Mixed Equations: Convert the fully known side to Base 10 first, then expand and match coefficients.

Final Checklist: Two's Complement Arithmetic

Final Checklist Subtraction: A - B = A + (2’s comp of B) . Carry-out: Always discard in signed addition. Overflow Condition 1: Operands same sign, Result different sign. Overflow Condition 2: V = C_{in} \oplus C_{o u t} at MSB. Shortcut: Unsigned overflow = Carry-out. Signed overflow = XOR of carries.

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Question 1

Level 1: Warm-up

Assertion (A): In a 4-bit 2's complement system, adding (- 1) and (- 2) does not result in an overflow. Reason (R): The addition produces a carry out of 1 from the most significant bit, which indicates an overflow in signed arithmetic.

Question 2

Level 1: Warm-up

Which of the following statements correctly describes the hardware operation for subtracting an n -bit number B from A in 2's complement arithmetic?

Question 3

Level 1: Warm-up

When converting the base-4 number (132)_{4} to base 5, the process involves an intermediate conversion to base 10. How many division steps are required to convert this base-10 value into base 5?

Question 4

Level 1: Warm-up

When converting the base-4 number (102)_{4} to base 5, the process requires routing through base 10. What is the minimum number of division-by-5 steps required to complete the base-10 to base-5 conversion?

Question 5

Level 1: Warm-up

When converting a decimal number to base r using the repeated division method, what is the maximum possible value that any remainder can take?

Question 6

Level 1: Warm-up

Which of the following statements is TRUE regarding the conversion of a number from base 3 to base 7?

Question 7

Level 1: Warm-up

When converting a base-10 integer to base 8 using the repeated division method, what is the maximum possible remainder that can be recorded in any single division step?

Question 8

Level 1: Warm-up

Which of the following statements correctly describes the 'Golden Rule' for converting a number from base 6 to base 9?

Question 9

Level 1: Warm-up

What is the maximum decimal value that can be represented by a 4-bit binary fraction of the form ?

Question 10

Level 1: Warm-up

When converting the decimal fraction to binary using the repeated multiplication method, what is the minimum number of multiplication steps required until the fractional part becomes exactly zero?

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Number Systems, Binary Arithmetic and Data Representation Short Notes for GATE CS

Number Systems, Binary Arithmetic and Data Representation short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Summary: The Radix Conversion Checklist

Revision Checklist

1. Validity Check: Are all digits $< r$ ?
2. Identify Bases: Source and Target?
3. Power-of-2? Use bit grouping (3 or 4 bits).
4. Otherwise: Route through Base 10.
5. Unknown $r$ : Solve polynomial, verify $r > max (d_{i})$ .

Summary: Fractional Conversion Checklist

Final Revision Checklist

1 Radix Point: Integer weights are positive powers (right to left); fractional weights are negative powers (left to right).
2 Binary → Decimal: Sum $d_{i} \times 2^{- i}$ . Rely on memorized values: $0.5, 0.25, 0.125, 0.0625$ .
3 Decimal → Binary: Repeated multiply by 2. Read integer parts top-down.
4 Non-Terminating: If it doesn't terminate, check for bit-limits or closest approximations.
5 Mixed Equations: Convert the fully known side to Base 10 first, then expand and match coefficients.

Final Checklist: Two's Complement Arithmetic

Final Checklist

Subtraction: $A - B = A + (2’s comp of B)$ .
Carry-out: Always discard in signed addition.
Overflow Condition 1: Operands same sign, Result different sign.
Overflow Condition 2: $V = C_{in} \oplus C_{o u t}$ at MSB.
Shortcut: Unsigned overflow = Carry-out. Signed overflow = XOR of carries.

Number Systems, Binary Arithmetic and Data Representation: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Digital Logic MCQ

Assertion (A): In a 4-bit 2's complement system, adding $(- 1)$ and $(- 2)$ does not result in an overflow.

Reason (R): The addition produces a carry out of 1 from the most significant bit, which indicates an overflow in signed arithmetic.

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is NOT the correct explanation of A.
C.
A is true, but R is false.
D.
A is false, but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: This is a construction question testing the precise definition of overflow and the common trap of confusing carry-out with overflow.

Step 1: Evaluate Assertion (A). $(- 1)_{10} = 111 1_{2}$ , $(- 2)_{10} = 111 0_{2}$ .

Step 2: Add them: $1111 + 1110 = 11101$ . Discard the carry out. The 4-bit result is $110 1_{2}$ , which is $(- 3)_{10}$ .

Step 3: Since $- 3$ is within the 4-bit range $[- 8, 7]$ , no overflow occurred. Assertion (A) is TRUE.

Step 4: Evaluate Reason (R). The addition does produce a carry out of 1. However, in signed 2's complement arithmetic, a carry out does NOT indicate overflow. Overflow is determined by the sign bits or XOR of carries. Thus, Reason (R) is FALSE.

Answer: A is true, but R is false.

Question 2 · Digital Logic MCQ

Which of the following statements correctly describes the hardware operation for subtracting an $n$ -bit number $B$ from $A$ in 2's complement arithmetic?

A.
Add $A$ and $B$ , then invert the result.
B.
Add $A$ to the 1's complement of $B$ and keep the carry out.
C.
Add $A$ to the 2's complement of $B$ and discard any carry out of the sign bit.
D.
Subtract $B$ from $A$ bit-by-bit and borrow from the sign bit.

Correct Answer:

Step-by-Step Solution

Key idea: This is a bounding question testing the constraints of unified hardware logic for subtraction in 2's complement.

Step 1: Recall that subtraction $A - B$ is performed by adding $A$ to the 2's complement of $B$ .

Step 2: The 2's complement of $B$ is found by inverting the bits of $B$ and adding 1.

Step 3: After the addition is complete, any carry out of the most significant bit (sign bit) must be discarded in 2's complement arithmetic.

Step 4: Evaluate the options. Option C perfectly matches this exact procedure.

Answer: Add $A$ to the 2's complement of $B$ and discard any carry out of the sign bit.

Question 3 · Digital Logic MCQ

When converting the base-4 number $(132)_{4}$ to base 5, the process involves an intermediate conversion to base 10. How many division steps are required to convert this base-10 value into base 5?

A.
2
B.
3
C.
4
D.
5

Correct Answer:

Step-by-Step Solution

Key idea: This is a step-counting question based on the cross-base conversion method.

Step 1: Convert $(132)_{4}$ to base 10.

$(1 \times 4^{2}) + (3 \times 4^{1}) + (2 \times 4^{0}) = 16 + 12 + 2 = 30$ .

Step 2: Convert 30 to base 5 using repeated division.

Step 3: Count the division steps until the quotient is 0.

Step 1: $30 \div 5 = 6$ , remainder 0.
Step 2: $6 \div 5 = 1$ , remainder 1.
Step 3: $1 \div 5 = 0$ , remainder 1.

Step 4: The quotient is now 0, so we stop. There are exactly 3 division steps.

Answer: 3.

Question 4 · Digital Logic MCQ

When converting the base-4 number $(102)_{4}$ to base 5, the process requires routing through base 10. What is the minimum number of division-by-5 steps required to complete the base-10 to base-5 conversion?

A.
1
B.
2
C.
3
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction question that requires performing the conversion and carefully counting the steps to avoid overcounting.

Step 1: Convert $(102)_{4}$ to base 10.

$(1 \times 4^{2}) + (0 \times 4^{1}) + (2 \times 4^{0}) = 16 + 0 + 2 = 18$ .

Step 2: Convert 18 to base 5 using repeated division.

Step 3: Count the division steps until the quotient is 0.

Step 1: $18 \div 5 = 3$ , remainder 3.
Step 2: $3 \div 5 = 0$ , remainder 3.

Step 4: The quotient is now 0, so we stop. There are exactly 2 division steps.

Answer: 2.

Question 5 · Digital Logic MCQ

When converting a decimal number to base $r$ using the repeated division method, what is the maximum possible value that any remainder can take?

A.
$r$
B.
$r - 1$
C.
$r + 1$
D.
$2 r$

Correct Answer:

Step-by-Step Solution

Key idea: This is a boundary condition question about the division algorithm.

Step 1: Recall the division algorithm: Dividend = Divisor $\times$ Quotient + Remainder.

Step 2: By definition, the remainder must always be strictly less than the divisor. If the remainder were equal to or greater than the divisor, you could divide again.

Step 3: Since the divisor in this method is the target base $r$ , the remainder must be less than $r$ .

Step 4: Since remainders are non-negative integers, the maximum possible value is $r - 1$ .

Answer: $r - 1$ .

Question 6 · Digital Logic MCQ

Which of the following statements is TRUE regarding the conversion of a number from base 3 to base 7?

A.
It can be performed directly by grouping bits into chunks of 3.
B.
It must be routed through base 10 as an intermediate step.
C.
It can be performed directly by grouping bits into chunks of 7.
D.
It requires conversion to base 2 before converting to base 7.

Correct Answer:

Step-by-Step Solution

Key idea: This is a statement evaluation question testing the 'Golden Rule' of cross-base conversion.

Step 1: Recall the rule for direct conversion. Direct bit-grouping only works when both bases are powers of the same number (e.g., 2, 4, 8, 16).

Step 2: Check the bases in the question: base 3 and base 7. Neither is a power of 2, and they do not share a common power base.

Step 3: Apply the Golden Rule. For arbitrary bases that are not powers of each other, you must route the conversion through base 10.

Step 4: Evaluate options. Options A and C suggest direct grouping, which is invalid here. Option D suggests routing through base 2, which is only useful if the bases were powers of 2. Option B correctly states the Golden Rule.

Answer: It must be routed through base 10 as an intermediate step.

Question 7 · Digital Logic MCQ

When converting a base-10 integer to base 8 using the repeated division method, what is the maximum possible remainder that can be recorded in any single division step?

A.
6
B.
7
C.
8
D.
9

Correct Answer:

Step-by-Step Solution

Key idea: This is an observation question about the boundary conditions of the division algorithm.

Step 1: Recall the division algorithm: Dividend = Divisor $\times$ Quotient + Remainder.

Step 2: The remainder must always be strictly less than the divisor. If the remainder were equal to or greater than the divisor, the quotient would increase.

Step 3: Here, the divisor is the target base, which is 8.

Step 4: Therefore, the remainder must be strictly less than 8. The maximum integer less than 8 is 7.

Answer: 7.

Question 8 · Digital Logic MCQ

Which of the following statements correctly describes the 'Golden Rule' for converting a number from base 6 to base 9?

A.
Direct bit-grouping can be used by grouping digits into chunks of 3.
B.
The conversion must be routed through base 10 as an intermediate step.
C.
The conversion can be done directly by dividing the base-6 number by 9.
D.
The conversion must be routed through base 2 first, then to base 9.

Correct Answer:

Step-by-Step Solution

Key idea: This is a bounding question testing the constraints of the 'Golden Rule' of cross-base conversion.

Step 1: Recall the Golden Rule. Direct bit-grouping only works when both bases are powers of the same number (e.g., 2, 4, 8, 16).

Step 2: Check the bases in the question: base 6 and base 9. Neither is a power of 2, and they do not share a common power base.

Step 3: Apply the rule. For arbitrary bases that are not powers of each other, you must route the conversion through base 10.

Step 4: Evaluate options. Options A, C, and D suggest invalid direct methods. Option B correctly states the mandatory intermediate step.

Answer: The conversion must be routed through base 10 as an intermediate step.

Question 9 · Digital Logic MCQ

What is the maximum decimal value that can be represented by a 4-bit binary fraction of the form $(0. b_{1} b_{2} b_{3} b_{4})_{2}$ ?

A.
0.9375
B.
0.8750
C.
0.9999
D.
0.5000

Correct Answer:

Step-by-Step Solution

Key idea: This is an observation question about the boundary of fractional representation using memorized weights.

Step 1: The maximum value occurs when all bits $b_{1}, b_{2}, b_{3}, b_{4}$ are set to 1.

Step 2: The value is the sum of the first four negative powers of 2: $2^{- 1} + 2^{- 2} + 2^{- 3} + 2^{- 4}$ .

Step 3: Using memorized weights: $0.5 + 0.25 + 0.125 + 0.0625$ .

Step 4: Summing these gives exactly $0.9375$ .

Answer: 0.9375.

Question 10 · Digital Logic MCQ

When converting the decimal fraction $(0.625)_{10}$ to binary using the repeated multiplication method, what is the minimum number of multiplication steps required until the fractional part becomes exactly zero?

A.
2
B.
3
C.
4
D.
5

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction question where the process seems like it might repeat infinitely, but terminates cleanly if counted correctly without overcounting the final zero state.

Step 1: Multiply $0.625 \times 2 = 1.25$ . Integer part is 1, new fraction is $0.25$ . (Step 1)

Step 2: Multiply $0.25 \times 2 = 0.50$ . Integer part is 0, new fraction is $0.50$ . (Step 2)

Step 3: Multiply $0.50 \times 2 = 1.00$ . Integer part is 1, new fraction is $0.00$ . (Step 3)

Step 4: The fractional part is now exactly zero, so the process terminates. There are exactly 3 multiplication steps.

Answer: 3.

number systems binary arithmetic and data representation short notes

Summary: The Radix Conversion Checklist

Revision Checklist

Summary: Fractional Conversion Checklist

Final Revision Checklist

Final Checklist: Two's Complement Arithmetic

Final Checklist

Try a question

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Number Systems, Binary Arithmetic and Data Representation Short Notes for GATE CS

Summary: The Radix Conversion Checklist

Revision Checklist

Summary: Fractional Conversion Checklist

Final Revision Checklist

Final Checklist: Two's Complement Arithmetic

Final Checklist

Number Systems, Binary Arithmetic and Data Representation: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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