Worksheet

A7.2 Binary arithmetic and signed integers

Data representation · A level · OCR H446 1.4.1, AQA 7517 4.5.2.1, Eduqas A500QS 2.3 · about 30 min

What this lesson is about

Unsigned addition and multiplication, sign and magnitude, two's complement, subtraction and overflow.

• Unsigned binary
• Sign and magnitude
• Two's complement
• Carry is not overflow
• Hexadecimal and signed values

Questions 6 marks in all

1. [1 mark]What is the denary value of the 8-bit two's complement number 11101100?

2. [1 mark]Write -37 as an 8-bit two's complement binary number.

3. [1 mark]What is the range of an 8-bit two's complement integer?

   1. A-128 to 127
   2. B-127 to 127
   3. C0 to 255
   4. D-255 to 255

4. [1 mark]Write -5 in 8-bit sign and magnitude.

5. [1 mark]01100100 + 01000110 is calculated in 8-bit two's complement. What happens?

   1. AOverflow: two positive numbers give a result with the sign bit set
   2. BA carry out of the top bit, which is an error
   3. CThe correct answer, 170
   4. DNothing unusual: the result is positive

6. [1 mark]What does this program print?

   a = 0b11111011
   b = 0b00000101
   total = a + b
   print(format(total & 0xFF, "08b"), total >> 8)

The task: a signed adder

Write add8(a, b). Each parameter is a string of exactly 8 characters, each 0 or 1, holding an 8-bit two's complement number. It returns a tuple (total, carry, overflow): - total: the 8-character bit string of the sum, with any carry out of the leftmost column dropped - carry: the carry out of the leftmost column, the integer 0 or 1 - overflow: True when both inputs have the same sign bit and total has the other one, otherwise False Add column by column from the right; do not convert to denary with int(..., 2), bin or format. The loop prints each line as 01110000 + 00110000 = 10100000 carry 0 overflow True.

# the two lines every program starts with: the commands, then the robot
from bugbot import *
connect()

def add8(a, b):
    total = ""
    carry = 0
    # add the columns from right to left here
    return total, carry, False

tests = [("01110000", "00110000"), ("11111011", "00000101"),
         ("10000000", "11111111"), ("00101101", "11110110")]
for a, b in tests:
    total, carry, overflow = add8(a, b)
    print(a, "+", b, "=", total, "carry", carry, "overflow", overflow)

Plan your program here, then type it in and press Run.

Do it on the robot
www.bugbotlab.com/learn/a7-2-binary-arithmetic-and-signed-integers/
The simulator checks it and tells you when it passes. Nothing to install, no account.

Challenges

1. Write sub8(a, b) that uses add8 and your own negate function, and test 50 - 25 and -100 - 100.
2. Write mul8(a, b) for unsigned bytes by shift and add. When should it report overflow?
3. Why can a positive number added to a negative number never overflow?