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In this section we will draw on what we have covered in this chapter and the one before, to present two small but complete programs to help consolidate what we have learned so far The rst program is a bit mathematical, but it is quite short at around 35 lines The second is concerned with text processing and is more substantial, with seven functions in around 80 lines of code
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Quadratic equations are equations of the form ax2 + bx + c = 0 where a 0 describe parabolas The roots of such equations are derived from the formula
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It is possible to use other encodings See the Python Tutorial s Source Code Encoding topic
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Examples
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b2 2
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x = b 2a 4ac The b 4ac part of the formula is called the discriminant if it is positive there are two real roots, if it is zero there is one real root, and if it is negative there are two complex roots We will write a program that accepts the a, b, and c factors from the user (with the b and c factors allowed to be 0), and then calculates and outputs the root or roots First we will look at a sample run, and then we will review the code
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quadraticpy ax + bx + c = 0 enter a: 25 enter b: 0 enter c: -725 25x + 00x + -725 = 0 x = 170293863659 or x = -170293863659
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With factors 15, -3, and 6, the output (with some digits trimmed) is:
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15x + -30x + 60 = 0 x = (1+17320508j) or x = (1-17320508j)
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The output isn t quite as tidy as we d like for example, rather than + -30x it would be nicer to have - 30x, and we would prefer not to have any 0 factors shown at all You will get the chance to x these problems in the exercises Now we will turn to the code, which begins with three imports:
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import cmath import math import sys
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We need both the float and the complex math libraries since the square root functions for real and complex numbers are different, and we need sys for sysfloat_infoepsilon which we need to compare oating-point numbers with 0 We also need a function that can get a oating-point number from the user:
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def get_float(msg, allow_zero): x = None while x is None: try: x = float(input(msg)) if not allow_zero and abs(x) < sysfloat_infoepsilon: print("zero is not allowed") x = None
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Since the Windows console has poor UTF-8 support, there are problems with a couple of the characters ( and ) that quadraticpy uses We have provided quadratic_unipy which displays the correct symbols on Linux and Mac OS X, and alternatives (^2 and ->) on Windows
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except ValueError as err: print(err) return x
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2 Data Types
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This function will loop until the user enters a valid oating-point number (such as 05, -9, 21, 492), and will accept 0 only if allow_zero is True Once the get_float() function is de ned, the rest of the code is executed We ll look at it in three parts, starting with the user interaction:
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print("ax\N{SUPERSCRIPT a = get_float("enter a: b = get_float("enter b: c = get_float("enter c: TWO} + bx + c = 0") ", False) ", True) ", True)
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Thanks to the get_float() function, getting the a, b, and c factors is simple The Boolean second argument says whether 0 is acceptable
x1 = None x2 = None discriminant = (b ** 2) - (4 * a * c) if discriminant == 0: x1 = -(b / (2 * a)) else: if discriminant > 0: root = mathsqrt(discriminant) else: # discriminant < 0 root = cmathsqrt(discriminant) x1 = (-b + root) / (2 * a) x2 = (-b - root) / (2 * a)
The code looks a bit different to the formula because we begin by calculating the discriminant If the discriminant is 0, we know that we have one real solution and so we calculate it directly Otherwise, we take the real or complex square root of the discriminant and calculate the two roots
equation = ("{0}x\N{SUPERSCRIPT TWO} + {1}x + {2} = 0" " \N{RIGHTWARDS ARROW} x = {3}")format(a, b, c, x1) if x2 is not None: equation += " or x = {0}"format(x2) print(equation)
We haven t done any fancy formatting since Python s defaults for oating-point numbers are ne for this example, but we have used Unicode character names for a couple of special characters