P3-M 4/21 Binary Overview
A series of binary lessons focusssed on math and conversions.
How to contact us
Join the "coding" channel on slack! That is the only place where we will be answering questions or sending announcements about lessons. If you have a question please contact us there.
How to join
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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?
There are 24 bits in 3 bytes.
- What digital information can be represented by bits?
Data can be represented with bits, which can include strings, integers, booleans, and images.
- Are bits an analog or digital form of storing data? What is the difference between the two?
Bits are digital because they are in the form og 0s and 1s. This can only be one of the other.
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?
The largest number that can be represented by 5 bits is 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?
The second language can represent 40 times more bits, which is 2^40 times as many bits as the first language.
- 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)
The largest number that can be used is 15 because 4 bits are reserved for representing negatives and positives and then the rest are reserved for representing actual numbers.
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(5, 2)
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 each 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?
The values of 0 to 4 are able to be represented by Base 5.
- What base is Hexadecimal? What range of values can each digit of Hexadecimal represent?
Hexadecimal is Base64, which can represent from 0 to f.
- 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?
In the Base30 system, the 23rd number would be n. For Base50 is would be n again.
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 = 2^3 + 2^1 = 8 + 2 = 10
- 11100010 = 2^7 + 2^6 + 2^5 + 2^1 = 128 + 64 + 32 + 2 = 226
- 10 = 2
-
Fill in the blanks (convert to binary)
- 12 = 2^3 + 2^2 = 1100
- 35 = 2^5 + 2^1 + 2^0 = 100011
- 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]
init = input("Would you like to add, subtract, multiply, or divide? (Input: a, s, m, or d)")
first = input("Input your first number:")
second = input("Input your first number:")
result = 0
decimal = 0
def bin_dec(binary):
decimal = 0
binary = str(binary)
for digit in binary:
decimal = decimal*2 + int(digit)
return decimal
if str(init) == "a":
num1 = int(bin_dec(first))
num2 = int(bin_dec(second))
result = num1 + num2
binnum1 = first
binnum2 = second
binary = bin(result)[2:]
decimal = bin_dec(binary)
print("Binary: ", binnum1, " + ", binnum2, " = ", binary)
print("Decimal: ", num1, " + ", num2, " = ", decimal)
elif str(init) == "s":
num1 = int(bin_dec(first))
num2 = int(bin_dec(second))
result = num1 - num2
binnum1 = first
binnum2 = second
binary = bin(result)[2:]
decimal = bin_dec(binary)
print("Binary: ", binnum1, " - ", binnum2, " = ", binary)
print("Decimal: ", num1, " - ", num2, " = ", decimal)
elif str(init) == "m":
num1 = int(bin_dec(first))
num2 = int(bin_dec(second))
result = num1 * num2
binnum1 = first
binnum2 = second
binary = bin(result)[2:]
decimal = bin_dec(binary)
print("Binary: ", binnum1, " * ", binnum2, " = ", binary)
print("Decimal: ", num1, " * ", num2, " = ", decimal)
elif str(init) == "d":
num1 = int(bin_dec(first))
num2 = int(bin_dec(second))
result = num1 / num2
binnum1 = first
binnum2 = second
binary = (bin(result)[2:])
decimal = bin_dec(binary)
print("Binary: ", binnum1, " / ", binnum2, " = ", binary)
print("Decimal: ", num1, " / ", num2, " = ", decimal)