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‎notebooks/algebra.ipynb

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"version": "3.11.5"
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‎notebooks/background.ipynb

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"- It is easier to learn than most programming languages.\n",
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"- It is a widespread language, available on almost all types of computers.\n",
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"- It has wide support for many common problems, and resources to learn how to solve them.\n",
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"- It is easier to run and to develop programs than many languages.\n"
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"- It is easier to run and to develop programs than many languages.\n",
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"\n",
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"<a id=\"mathematical-basics\"></a>\n",
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"# Mathematical basics\n",
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"You must understand various mathematical concepts to program.\n",
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"These sections include some programming specific information you may not have seen,\n",
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"you might take a look even if you are familiar with them.\n",
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"These include:\n",
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"\n",
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"|Concept |Description |\n",
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"|:- |:- |\n",
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"|[Arithmetic](./arithmetic.ipynb) |Basic operations and properties of numbers |\n",
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"|[Algebra](./algebra.ipynb) |Solves problems using mathematical expressions |\n",
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"|[Number Systems](./number-systems.ipynb)|Representing numbers as numerals |"
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]
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{

‎notebooks/data.ipynb

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"Instructions can use literal constants or can use the values stored in variables. \n",
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"Values can be combined it collections such as lists and dictionaries.\n",
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"\n",
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"In Python, the data type is set when you assign a value to a variable:\n",
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"x = \"Hello World\"\tstr\t\n",
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"x = 20\tint\t\n",
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"x = 20.5\tfloat\t\n",
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"x = [\"apple\", \"banana\", \"cherry\"]\tlist\t\n",
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"x = range(6)\trange\t\n",
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"x = {\"name\" : \"John\", \"age\" : 36}\tdict\n",
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"In Python, the data type is set when you assign a value to a variable.\n",
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"\n",
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"statement|type\n",
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":- |:-\n",
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"x = \"Hello World\"|string\t\n",
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"x = 20|int\t\n",
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"x = 20.5|float\t\n",
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"x = [\"apple\", \"banana\", \"cherry\"]|list\t\n",
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"x = {\"name\" : \"John\", \"age\" : 36}|dictionary\n",
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"\n",
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"[Numbers](./numbers.ipynb) \n",
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"[Strings](./strings.ipynb) \n",

‎notebooks/number-systems.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "213ad744",
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"metadata": {},
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"source": [
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"# Number systems\n",
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"Number systems use symbols to represent numbers in a compact way.\n",
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"\n",
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"[Additive number systems](#Additive-number-systems) \n",
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"[Exponents](#Exponents) \n",
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"[Positional number systems](#Positional-number-systems) \n",
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"[Binary numbers](#Binary-numbers) \n",
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"[Operations on binary numbers](#Operations-on-binary-numbers) \n",
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"[Boolean logic](#Boolean-logic)\n",
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"\n",
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"\n",
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"<a id=\"Additive-number-systems\"></a>\n",
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"## Additive number systems\n",
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"\n",
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"The original way of counting, once the obvious limits of counting on fingers was reached, was using *tally marks* or notches on a stick, stone, or piece of clay as symbols each representing a single thing.\n",
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"These were added to together to create a *number*, which symbolizes a quantity of things.\n",
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"Though this was less limited than using fingers, these marks likewise reached a limit as numbers increased.\n",
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"\n",
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"The Roman approach to this was to use symbols or *numerals* to count groups, then these numerals were added together to get numbers.\n",
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"\n",
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"|Number |1 |2 |3 |4 |5 |9 |10 |50 |100 |500 |1000\n",
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"|:- |:- |:- |:- |:- |:- |:- |:- |:- |:- |:- |:-\n",
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"|Roman |I |II |III |IV |V |IX |X |L |C |D |M\n",
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"\n",
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"Larger numbers could be represented using fewer symbols. \n",
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"This is an example of adding these numberals together.\n",
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"\n",
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"DCCXIV is 500 + 200 + 10 + 4 or 714.\n",
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"\n",
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"Very large numbers still need quite a few symbols. \n",
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"Adding numbers is straightforward, but becomes cumbersome.\n",
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"Substracting numbers is a little harder, and multiplying and dividing numbers is harder yet.\n",
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"\n",
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"In India, a new number system, the *Hindu* system, was created that used fewer symbols but could represent large numbers in a more compact way. Some background is needed to discuss this system.\n",
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"\n",
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"<a id=\"Positional-number-systems\"></a>\n",
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"## Positional number systems\n",
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"\n",
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"Positional number systems use fewer numerals than additive number systems, \n",
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"but the numerals have different values depending on their *position* in a number.\n",
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"The Hindu number system uses the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. \n",
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"We inherited the number system from the Arabic cultures, and the number system uses Hindu-Arabic numerals or what we call Arabic numerals.\n",
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"The number system is called *decimal* because it uses 10 different numerals, each of which we call *digits*.\n",
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"We use [*exponents*](./Arithmetic.ipynb#Exponents) to show the different powers of 10 multiplied by each digit in different positions, so \n",
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"\n",
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"$123 = 100 + 20 + 3 = 1 \\cdot 100 + 2 \\cdot 10 + 3 \\cdot 1 = 1 \\cdot 10^2 + 2 \\cdot 10^1 + 3 \\cdot 10^0$,\n",
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"\n",
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"|position |3 |2 |1\n",
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"|:-|:-|:-|:-\n",
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"|digit |1 | 2 | 3\n",
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"|multiplier |100 |10 |1\n",
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"|using exponents |$10^2$ |$10^1$ |$10^0$ \n",
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"|value |100 |20 |3\n",
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"\n",
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"To decide what digit goes in a position that has no value, the Hindu system had to invent the idea of a zero, the thing that is nothing and if multiplied by any number gives nothing. \n",
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"The concept of zero was new because it was not used in additive systems. \n",
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"As an example, the number 305 needs zero in the second position to work, so that we have \n",
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"\n",
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"3 x 100 + 0 x 10 + 5 x 1 = 300 + 0 + 5 = 305\n",
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"\n",
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"digit |3 |0 |5\n",
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"|:- |:- |:- |:-\n",
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"position |3 |2 |1\n",
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"multiplier |100 |10 |1\n",
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"value |3 x 100 = 300 |0 x 10 = 0 |5 x 1 = 5\n",
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"\n",
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"In the decimal system, adding numbers is simplified, you add each position separately.\n",
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"There are fewer numerals than in the Roman system, so addition is faster.\n",
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"For 305 + 262 = 567\n",
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"\n",
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"position |1 |2 |3\n",
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"|:- |:- |:- |:-\n",
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"| |3 |0 |5\n",
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"|+ |2 |6 |2\n",
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"| |--- |--- |---\n",
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"| |5 |6 |7\n",
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"\n",
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"When adding numbers in a position gives more than 9, we introduce the idea of a *carry*.\n",
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"For 7 + 35 = 42\n",
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"\n",
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"|position |1 |2\n",
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"|:- |:- |:- \n",
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"|carry |1 |\n",
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"| | |7\n",
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"|+ |3 |5\n",
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"| |--- |--- |---\n",
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"| |4 |2\n",
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"\n",
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"Subtraction, multiplication, and division are similarly simplified.\n",
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"\n",
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"<a id=\"Binary-numbers\"></a>\n",
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"## Binary numbers\n",
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"In the decimal system, we use a base of 10 for the exponents at each numeral position and the digits 0 through 9.\n",
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"There are as many different digits as the number base.\n",
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"There is no reason why we can't use another number for the base, such as 2.\n",
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"We call a base 2 number system a *binary* system.\n",
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"There are two numerals in a binary system, 0 and 1.\n",
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"We can give the value for a binary number 10110, which we show as $10110_2$ to indicate it is base 2, as\n",
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"\n",
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"digit |1 |0 |1 |1 |0\n",
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":- |:- |:- |:- |:- |:-\n",
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"position |5 |4 |3 |2 |1\n",
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"multiplier |16 |8 |4 |2 |1\n",
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"using exponents |$2^4$ |$2^3$ |$2^2$ |$2^1$ |$2^0$\n",
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"position value |16 |0 |4 |2 |0\n",
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"\n",
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"so the final number is 16 + 0 + 4 + 2 + 0 = 22. \n",
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"We can show decimal 22 as $22_{10}$ to show the number base.\n",
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"\n",
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"We call each binary numeral a *binary digit* or *bit*.\n",
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"Computers are able to represent two values, 0 and 1, well. \n",
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"In electronic circuits 0 corresponds to \"no power\" or \"no voltage\" and 1 corresponds to \"positive voltage\".\n",
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"It turns out trying to electrically detect 10 voltage levels for 10 different decimal digits is unreliable.\n",
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"\n",
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"We can add numbers like we do in the decimal system. We add each position separately.\n",
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"\n",
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"$101_2 + 10_2 = 111_2$\n",
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"\n",
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"position |1 |2 |3\n",
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"|:- |:- |:- |:-\n",
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"| |1 |0 |1\n",
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"|+ | |1 |0\n",
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"| |--- |--- |---\n",
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"| |1 |1 |1\n",
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"\n",
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"When adding numbers in a position gives more than 1, we introduce the idea of a *carry*.\n",
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"\n",
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"$101_2 + 1_2 = 110_2$\n",
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"\n",
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"|position |1 |2 |3\n",
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"|:- |:- |:- |:- \n",
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"|carry | |1 |\n",
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"| | 1| 0| 1\n",
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"|+ | | |1\n",
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"| |--- |--- |---\n",
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"| |1 |1 | 0\n",
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"\n",
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"Subtraction, multiplication, and division are similarly simplified.\n",
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"\n",
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"Using binary numbers can also get tedious.\n",
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"A shorthand for binary numbers is using base 16 numbers, or *hexadecimal* numbers.\n",
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"Hexadecimal numbers use 16 numerals.\n",
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"We can use decimal digits for the first 10, 0 through 9, but we need different numberals for the last 6.\n",
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"We use the six letters A through F for them.\n",
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"We typically put '0x' in from on the number to show it is hexademical.\n",
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"\n",
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"|decimal |binary |hexadecimal\n",
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"|:-|:-|:- \n",
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"|0 |$0_2$ |0x0|\n",
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"|1 |$1_2$ |0x1 |\n",
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"|2 |$10_2$ |0x2 |\n",
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"|3 |$11_2$ |0x3 |\n",
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"|4 |$100_2$ |0x4 |\n",
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"|5 |$101_2$ |0x5 |\n",
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"|6 |$110_2$ |0x6 |\n",
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"|7 |$111_2$ |0x7 |\n",
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"|8 |$1000_2$ |0x8 |\n",
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"|9 |$1001_2$ |0x9 |\n",
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"|10 |$1010_2$ |0xA |\n",
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"|11 |$1011_2$ |0xB |\n",
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"|12 |$1100_2$ |0xC |\n",
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"|13 |$1101_2$ |0xD |\n",
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"|14 |$1110_2$ |0xE |\n",
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"|15 |$1111_2$ |0xF |\n",
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"\n",
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"In computers, only a fixed number of bits are allocated for numbers.\n",
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"This limits the size of number that can be stored.\n",
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"Typically sizes for numbers are 8, 16, 32, or 64 bits.\n",
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"The term for an 8 bit number is a *byte*.\n",
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"We can find the maximum size for a number from the value of a number with all 1 bits.\n",
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"The largest number that can be stored in 8 bits is\n",
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"\n",
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"$11111111_2 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255$\n",
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"\n",
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"This value is $2^8 - 1$. The names and maximum values for different types of numbers of different sizes are given by\n",
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"\n",
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"|size |bits |bytes |maximum value\n",
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"|:- |:- |:- |:- \n",
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"|byte |8 |1 |$2^8 - 1$ |\n",
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"|short |16 |1 |$2^{16} - 1$ |\n",
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"|int |32 |1 |$2^{32} - 1$ |\n",
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"|long |64 |1 |$2^{64} - 1$ |\n",
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"\n",
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"Numbers being limited by the number of bits requires that programs must anticipate the largest possible value and use a size large enough to store it.\n",
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"When a number too large for the number of bits is stored, we say an *overflow* occurs.\n",
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"In that case, the bits that cannot be stored are, for example, thrown away and the value is incorrect.\n",
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"\n",
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"<a id=\"Operations-on-binary-numbers\"></a>\n",
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"## Operations on binary numbers\n",
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"Adding binary numbers is done like in the decimal system.\n",
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"To add the binary numbers, we get\n",
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"\n",
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"$101_2 + 1_2 = 110_2$.\n",
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"\n",
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"position |1 |2 |3 |4\n",
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":- |:- |:- |:- |:- \n",
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"|carry | | |1 | | |\n",
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"| | |1 |0 |1 |\n",
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"|+ | | | |1\n",
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"| |--- |--- |--- |---\n",
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"| | |1 |1 |0\n",
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"\n",
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"Subtraction, multiplication, and division are similarly done as with decimal numbers. \n",
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"\n",
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"<a id=\"Boolean-logic\"></a>\n",
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"## Boolean logic\n",
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"The binary numerals 1 and 0 can conveniently represent truth values, true and false.\n",
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"A logic based on these values, *Boolean logic* (invented by mathematician George Boole),\n",
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"gives new operations on binary numbers.\n",
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"The basic operations are *AND*, *OR*, *NOT*, and *XOR*.\n",
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"\n",
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"|operation | description |example\n",
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"|:- |:- |:-\n",
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"|AND |result is true if each operand is true|1 AND 1 = true\n",
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"|OR |result is true if either operand is true|0 AND 1 = true\n",
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"|NOT |result is the opposite of the operand |NOT 1 = false\n",
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"|XOR |result is false if both operands are the same, and true if both are different|1 AND 1 = false\n",
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"\n",
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"The results of these operations can be shown in *truth tables*.\n",
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"This is the truth table for AND.\n",
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"\n",
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"|AND |1 |0\n",
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"|:-|:- |:- \n",
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"|1 |1 | 0\n",
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"|0 |0 | 0\n",
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"\n",
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"This is the truth table for OR.\n",
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"\n",
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"|OR |1 |0\n",
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"|:-|:- |:- \n",
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"|1 |1 | 1\n",
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"|0 |1 | 0\n",
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"\n",
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"This is the truth table for NOT.\n",
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"\n",
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"|NOT | |\n",
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"|:-|:- |\n",
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"|1 |0 |\n",
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"|0 |1 |\n",
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"\n",
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"This is the truth table for XOR.\n",
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"\n",
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"|XOR |1 |0\n",
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"|:-|:- |:- \n",
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"|1 |0 | 1\n",
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"|0 |1 | 0\n",
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"\n",
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"These operations are used in operations on binary numbers in programs."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "43d3987e",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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‎notebooks/numbers.ipynb

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"## Number operators\n",
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"The usual arithmetic operators can be used, like addition `+` and subtraction `-`. \n"
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"The usual [arithmetic operators](./arithmetic.ipynb/#Operations-on-numbers) can be used, like addition `+` and subtraction `-`. \n"
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"print(\"9 % 2 is \", x)"
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"Order of operations\n",
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"\n",
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"The precedence order is described in the table below, starting with the highest precedence at the top:\n",
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"\n",
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"Operator\tDescription\tTry it\n",
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"()\tParentheses\t\n",
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"**\tExponentiation\t\n",
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"+x -x ~x\tUnary plus, unary minus, and bitwise NOT\t\n",
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"* / // %\tMultiplication, division, floor division, and modulus\t\n",
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"+ -\tAddition and subtraction"
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]
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},
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"cell_type": "markdown",
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"id": "f299a8cc-64c6-4b13-86bd-85d607038c1a",

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