P3-M 4/21 Binary Overview
A series of binary lessons focusssed on math and conversions.
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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 bits
-
What digital information can be represented by bits?: booleans
-
Are bits an analog or digital form of storing data? What is the difference between the two?: Digital form of storing data, analog is physical, digital is virtual or digital.
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(100)
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?: 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?: 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): 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?): when the number for the power is set too high, the program overflows
exponent(5, 10000)
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?: 0,1,2,3,4
-
What base is Hexadecimal? What range of values can each digit of Hexadecimal represent?: Base 16: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,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?: Base 30 --> n, also Base 50 --> n
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 = 10_
- 11100010 = 226_
- 10 = 2_
-
Fill in the blanks (convert to binary)
- 12 = 001100_
- 35 = 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]
digit1 = input("Enter first binary digits: ")
digit2 = input("Enter second binary digits: ")
decimalResult = 0
binaryResult = 0
finish=0 # to track completion
# converts to binary from decimal
def toBinaryValue(num):
bin = 0
# array to store binary digits
arr = []
# loop till end of all digits
while num > 0:
# mod of 2 will give either 0 or 1
# append to array
arr.append(num % 2)
# divide the number by 2 to move to next digit
num //= 2
arr.reverse()
# convert to displayable format
for i in arr:
bin = bin * 10 + i
return bin
# converts to decimal from binary
def toDecimalValue(num):
pos = []
index = 0
decimal = 0
# store the number in a list array
numArray = list(str(num))
#reverse the number
numArray.reverse()
# loop till the length of the digit and
# check for bit 1 or bit 0
for x in numArray:
if (x == "1"):
# if bit 1 then store the index in pos array
pos.append(index)
index += 1
else:
# if bit 0 then pos array will be zero value
index +=1
# loop the pos array and raise it to the power of 2
for x in pos:
decimal += 2**x
return decimal
print("Number 1: Binary: " + digit1 + " Decimal: " + str(toDecimalValue(digit1)))
print("Number 2: Binary: " + digit2 + " Decimal: " + str(toDecimalValue(digit2)))
# Do Addition
decimalResult = toDecimalValue(digit1) + toDecimalValue(digit2)
binaryResult = toBinaryValue(int(decimalResult))
print("Addition: " + digit1, "+", digit2, "= Decimal value is:", decimalResult, "and Binary value is:", binaryResult)
# Do Subtraction
decimalResult = toDecimalValue(digit1) - toDecimalValue(digit2)
binaryResult = toBinaryValue(int(decimalResult))
print("Subtraction: " + digit1, "-", digit2, "= Decimal value is:", decimalResult, "and Binary value is:", binaryResult)
# Do Multiplication
decimalResult = toDecimalValue(digit1) * toDecimalValue(digit2)
binaryResult = toBinaryValue(int(decimalResult))
print("Multiplication: " + digit1, "*", digit2, "= Decimal value is:", decimalResult, "and Binary value is:", binaryResult)
# Do Division
decimalResult = toDecimalValue(digit1) / toDecimalValue(digit2)
binaryResult = toBinaryValue(int(decimalResult))
print("Division: " + digit1, "/", digit2, "= Decimal value is:", decimalResult, "and Binary value is:", binaryResult)