P3-M 4/21 Binary Overview
A series of binary lessons focusssed on math and conversions.
How to contact us
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- Click on "browse channels"
- Search for "coding"
- Click the green "Join" button on the right
Learning Objectives
DAT-1.A: Representing Data with Bits
Basic Information
- Bit is short for binary digit, and represents a value of either 0 or 1.
- A byte is 8 bits.
- Sequences of bits are used to represent different things.
- Representing data with sequences of bits is called abstraction.
Practice Questions:
- How many bits are in 3 bytes? 24
- What digital information can be represented by bits? yes/no, on/off, +/-...
- Are bits an analog or digital form of storing data? What is the difference between the two? analog; not continuous #### Examples
- Boolean variables (true or false) are the easiest way to visualize binary.
- 0 = False
- 1 = True
import random
def example(runs):
# Repeat code for the amount of runs given
while runs > 0:
# Assigns variable boolean to either True or False based on random binary number 0 or 1.
boolean = False if random.randint(0, 1) == 0 else True
# If the number was 1 (True), it prints "awesome."
if boolean:
print("binary is awesome")
# If the number was 2 (False), it prints "cool."
else:
print("binary is cool")
runs -= 1
# Change the parameter to how many times to run the function.
example(10)
DAT-1.B: The Consequences of Using Bits to Represent Data
Basic Information
- Integers are represented by a fixed number of bits, this limits the range of integer values. This limitation can result in overflow or other errors.
- Other programming languages allow for abstraction only limited by the computers memory.
- Fixed number of bits are used to represent real numbers/limits
Practice Questions:
- What is the largest number can be represented by 5 bits? 1+2+2^2+2^3+2^4+2^5 = 32-1=31
-
One programing language can only use 16 bits to represent non-negative numbers, while a second language uses 56 bits to represent numbers. How many times as many unique numbers can be represented by the second language? 56-16=40---> 2^40; 2^56/2^16= 2^40
-
5 bits are used to represent both positive and negative numbers, what is the largest number that can be represented by these bits? (hint: different thatn question 1)
- first bit will depresent whether the number is negative or positive
- so it would be the largest 4 bit number
- So the largest number that can be rep is 15
Examples
import math
def exponent(base, power):
# Print the operation performed, turning the parameters into strings to properly concatenate with the symbols "^" and "=".
print(str(base) + "^" + str(power) + " = " + str(math.pow(base, power)))
# How can function become a problem? (Hint: what happens if you set both base and power equal to high numbers?)
exponent(156, 257)
DAT-1.C: Binary Math
Basic Information
- Binary is Base 2, meaning each digit can only represent values of 0 and 1.
- Decimal is Base 10, meaning eacht digit can represent values from 0 to 9.
- Conversion between sequences of binary to decimal depend on how many binary numbers there are, their values and their positions.
Practice Questions:
- What values can each digit of a Base 5 system represent?
- 0-4
-
What base is Hexadecimal? What range of values can each digit of Hexadecimal represent? base 16.
- 0-15
-
When using a base above 10, letters can be used to represent numbers past 9. These letters start from A and continue onwards. For example, the decimal number 10 is represented by the letter A in Hexadecimal. What letter would be used to represent the Base 10 number 23 in a Base 30 system? What about in a Base 50 system?
- W, 23
Examples
- Using 6 bits, we can represent 64 numbers, from 0 to 63, as 2^6 = 64.
- The numbers in a sequence of binary go from right to left, increasing by powers of two from 0 to the total amount of bits. The whole number represented is the sum of these bits. For example:
- 111111
- 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0
- 32 + 16 + 8 + 4 + 2 + 1
- 63
-
Fill in the blanks (convert to decimal)
- 001010 = (2^1)+2^3 = 10
- 11100010 = 2^1+2^5+2^6+2^7= 226
- 10 = 2^1= 2
-
Fill in the blanks (convert to binary)
- 12 = 001100
-
35 = 100011
35/2= 17 +1 17/2 = 8 + 1 8/2 = 4 +0 4/2 = 2 +0 2/2 = 1 +0 1/2 = 0 + 1
-
256 = 100000000
Hacks & Grading (Due SUNDAY NIGHT 4/23)
- Complete all of the popcorn hacks (Fill in the blanks + run code cells and interact + Answer ALL questions) [0.3 or nothing]
- Create a program to conduct basic mathematical operations with binary sequences (addition, subtraction, multiplication, division) [0.6 or nothing]
- For bonus, program must be able to conduct mathematical operations on binary sequences of varying bits (for example: 101 + 1001 would return decimal 14.) [0.1 or nothing]
#decimal input---> binary and decimal output
a = int(input("Enter first number"))
b = int(input(("Enter second number")))
operation = input("Select operation, multiple (m), divide (d), add (a), or subtract (s)")
def decimaltobinary(num):
if num > 1:
decimaltobinary(num // 2) #divide input by 2
print(num % 2 , end = "")
def operate(a,b):
if operation == "m":
c = a * b
print (c)
print(decimaltobinary(c))
elif operation == "d":
c = a/b
print (c)
print(decimaltobinary(c))
elif operation == "a":
c = a + b
print (c)
print(decimaltobinary(c))
elif operation == "s":
c = a - b
print (c)
print(decimaltobinary(c))
operate(a,b)
a = input("Enter first binary number")
b = input(("Enter second binary number"))
operation = input("Select operation, multiple (m), divide (d), add (a), or subtract (s)")
def binarytodecimal(binary):
decimal = 0
for digit in binary:
decimal = decimal*2 + int(digit)
def decimaltobinary(num):
if num > 1:
decimaltobinary(num // 2) #divide input by 2
print(num % 2 , end = "")
def operateopposite(a,b):
if operation == "m":
c = binarytodecimal(a) * binarytodecimal(b)
print (c)
print(decimaltobinary(c))
elif operation == "d":
c = binarytodecimal(a) / binarytodecimal(b)
print (c)
print(decimaltobinary(c))
elif operation == "a":
c = binarytodecimal(a) + binarytodecimal(b)
print (c)
print(decimaltobinary(c))
elif operation == "s":
c = binarytodecimal(a) - binarytodecimal(b)
print (c)
print(decimaltobinary(c))
operateopposite(a,b)