To calculate Catalan numbers using binomial Coefficients, you first need to write a function that calculates binomial coefficients. exampledef binomialCoefficient(n, k):    # To optimize calculation of C(n, k)    if (k > n - k):       k = n - k    coeff = 1    for i in range(k):       coeff *= (n - i)       coeff /= (i + 1)    return coeff def catalan(n):    return binomialCoefficient(2*n, n) / (n + 1) for i in range (11):    print (catalan(i))OutputThis will give the output −1.0 1.0 2.0 5.0 14.0 42.0 132.0 429.0 1430.0 4862.0 16796.0

You can perform arithmetic operations on large numbers in python directly without worrying about speed. Python supports a "bignum" integer type which can work with arbitrarily large numbers. In Python 2.5+, this type is called long and is separate from the int type, but the interpreter will automatically use whichever is more appropriate.As long as you have version 2.5 or better, just perform standard math operations and any number which exceeds the boundaries of 32-bit math will be automatically (and transparently) converted to a bignum.examplea = 182841384165841685416854134135 b = 135481653441354138548413384135 print(a - b)OutputThis will give the output −47359730724487546868440750000

If you want something that converts integers to words like 99 to ninety-nine, you have to use an external package or build one yourself. The pynum2word module is pretty good at this task. You can install it using −$ pip install pynum2wordThen use it in the following way −>>> import num2word >>> num2word.to_card(16) 'sixteen' >>> num2word.to_card(23) 'twenty-three' >>> num2word.to_card(1223)'one thousand, two hundred and twenty-three'If you want to get results like 1.23 million for 1, 230, 000, you can use the humanize library to do so. You can install it using −$ pip install humanizeThen use it in the following way ... Read More

A perfect number is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3.You can find perfect numbers within a given range by testing each number for the given condition in the given range. exampledef print_perfect_nums(start, end):    for i in range(start, end + 1):    sum1 = 0    for x in range(1, i):       # Check if a divisor, if it is, add to sum       if(i % x == 0):          sum1 = sum1 + x          if (sum1 == i):             print(i) print_perfect_nums(1, 300)OutputThis will give the output6 28

You can format a floating number to a fixed width in Python using the format function on the string. examplenums = [0.555555555555, 1, 12.0542184, 5589.6654753] for x in nums:    print("{:10.4f}".format(x))OutputThis will give the output −0.5556 1.0000 12.0542 5589.6655ExampleUsing the same function, you can also format integers −nums = [5, 20, 500] for x in nums:    print("{:d}".format(x))OutputThis will give the output −5 20 500ExampleYou can use it to provide padding as well, by specifying the number before d:nums = [5, 20, 500] for x in nums:    print("{:4d}".format(x))OutputThis will give the output −5 20 500The https://pyformat.info/ website is a great ... Read More

If you have binary numbers as strings, you can convert them to ints first using int(str, base) by providing the base as 2. Then add the numbers like you'd normally do. Finally convert it back to a string using the bin function. For example,a = '001' b = '011' sm = int(a,2) + int(b,2) c = bin(sm) print(c)This will give the output:0b100

Python comes with an amazing standard math library that provides all trigonometric functions like sin, cos, etc. You can import that library and use these functions. Note that these functions expect angles in radians. For example,import math angle1 = math.radians(90) angle2 = math.radians(60) print(math.cos(angle1)) print(math.cos(angle2))This will give the output:6.123233995736766e-17 0.5The first value is very close to zero. Such errors come due to computation limits.

To check if a float value is a whole number, use the float.is_integer() method. For example,print((10.0).is_integer()) print((15.23).is_integer())This will give the outputTrue False

Objects of different types except numbers are ordered by their type names; objects of the same types that don’t support proper comparison are ordered by their address. When you order two strings or two numeric types the ordering is done in the expected way (lexicographic ordering for string, numeric ordering for integers).When you order a numeric and a non-numeric type, the numeric type comes first.If you have a number in a str object, you can simply convert it to a float or an int using their respective constructors. For example, i = 100 j = "12" int_j = int(j) print(int_j ... Read More

You can use relational operators in python to compare numbers(both float and int) in python. These operators compare the values on either side of them and decide the relation among them. Assume variable a holds 10 and variable b holds 20, thenOperatorExample==(a == b) is not true.!=(a != b) is true.>(a > b) is not true.=(a >= b) is not true.= b) print(a