From effb1e115a29d473297560f7328b01525835ac95 Mon Sep 17 00:00:00 2001 From: thomasabishop Date: Mon, 25 Apr 2022 07:42:49 +0100 Subject: [PATCH] Last Sync: 2022-04-25 07:42:49 --- .../Prealgebra/Add_Subtract_Fractions.md | 41 ++++++++++++++++++ Mathematics/Prealgebra/Dividing_fractions.md | 41 ++++++++++-------- .../Prealgebra/Handling_negative_fractions.md | 20 +++++---- .../Prealgebra/Least_common_denominator.md | 0 Mathematics/Prealgebra/Reciprocals.md | 9 ++-- pdf/Algorithmic_complexity.pdf | Bin 549310 -> 549310 bytes pdf/Arrays.pdf | Bin 83599 -> 83599 bytes pdf/Assembly_programs.pdf | Bin 58049 -> 58049 bytes pdf/Atomic_and_molecular_sentences.pdf | Bin 97737 -> 97737 bytes pdf/Axioms_of_set_theory.pdf | Bin 66749 -> 66749 bytes pdf/Basic_database_concepts.pdf | Bin 99218 -> 99218 bytes pdf/Basic_properties_of_sets.pdf | Bin 128524 -> 128524 bytes pdf/Biconditional_Elimination.pdf | Bin 39042 -> 39042 bytes pdf/Biconditional_Introduction.pdf | Bin 35537 -> 35537 bytes pdf/Binary_arithmetic.pdf | Bin 149838 -> 149838 bytes pdf/Binary_arithmetic_with_circuits.pdf | Bin 357156 -> 357156 bytes pdf/Bits,_bytes,_nibbles.pdf | Bin 51186 -> 51186 bytes pdf/Compile_from_source.pdf | Bin 49795 -> 49795 bytes pdf/Conditional_Elimination.pdf | Bin 43232 -> 43232 bytes pdf/Conditional_Introduction.pdf | Bin 74250 -> 74250 bytes pdf/Conjunction_Elimination.pdf | Bin 47565 -> 47565 bytes pdf/Conjunction_Introduction.pdf | Bin 45840 -> 45840 bytes pdf/Defining_a_computer.pdf | Bin 731550 -> 731550 bytes pdf/Design_features_of_language.pdf | Bin 60132 -> 60132 bytes pdf/Disjunction_Elimination.pdf | Bin 91007 -> 91007 bytes pdf/Disjunction_Introduction.pdf | Bin 54596 -> 54596 bytes pdf/HTTP_request_types.pdf | Bin 46765 -> 46765 bytes pdf/Indeterminacy.pdf | Bin 112534 -> 112534 bytes pdf/Law_of_the_Excluded_Middle.pdf | Bin 35622 -> 35622 bytes pdf/Logic_circuits.pdf | Bin 157687 -> 157687 bytes pdf/Logic_gates.pdf | Bin 407144 -> 407144 bytes pdf/Logical_possibility_and_necessity.pdf | Bin 122811 -> 122811 bytes pdf/Lovelace_quote.pdf | Bin 31770 -> 31770 bytes pdf/Morphology.pdf | Bin 77311 -> 77311 bytes pdf/Negation_Elimination.pdf | Bin 44476 -> 44476 bytes pdf/Negation_Introduction.pdf | Bin 44871 -> 44871 bytes pdf/Object_language_and_meta-language.pdf | Bin 131464 -> 131464 bytes pdf/Package_management_in_Arch.pdf | Bin 71061 -> 71061 bytes pdf/Primary_key.pdf | Bin 80548 -> 80548 bytes pdf/Queue.pdf | Bin 154497 -> 154497 bytes pdf/RESTful_APIs.pdf | Bin 105744 -> 105744 bytes pdf/Recursion.pdf | Bin 176160 -> 176160 bytes pdf/Reiteration.pdf | Bin 53262 -> 53262 bytes pdf/SQL_syntax.pdf | Bin 559941 -> 559941 bytes pdf/Semantic_versioning.pdf | Bin 118251 -> 118251 bytes pdf/Soundness.pdf | Bin 98882 -> 98882 bytes pdf/Stacks.pdf | Bin 151942 -> 151942 bytes pdf/Syllogism.pdf | Bin 79914 -> 79914 bytes pdf/The_Pragmatic_Programmer_1999.pdf | Bin 169339 -> 169339 bytes pdf/The_binary_number_system.pdf | Bin 146858 -> 146858 bytes pdf/Theorems_and_empty_sets.pdf | Bin 75643 -> 75643 bytes pdf/Turing_machines.pdf | Bin 102880 -> 102880 bytes pdf/User_management.pdf | Bin 74126 -> 74126 bytes pdf/Von_Neumann_architecture.pdf | Bin 239665 -> 239665 bytes pdf/Why_computers_use_binary.pdf | Bin 144674 -> 144674 bytes 55 files changed, 79 insertions(+), 32 deletions(-) create mode 100644 Mathematics/Prealgebra/Add_Subtract_Fractions.md create mode 100644 Mathematics/Prealgebra/Least_common_denominator.md diff --git a/Mathematics/Prealgebra/Add_Subtract_Fractions.md b/Mathematics/Prealgebra/Add_Subtract_Fractions.md new file mode 100644 index 0000000..8234a53 --- /dev/null +++ b/Mathematics/Prealgebra/Add_Subtract_Fractions.md @@ -0,0 +1,41 @@ +--- +tags: + - Mathematics + - Prealgebra + - fractions + - division +--- + +# Adding and subtracting fractions + +## Adding/ subracting fractions with common denominators + +For two fractions $\frac{a}{c}$ and $\frac{b}{c}$ with a common denominator, their sum is defined as: + +$$ + \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} +$$ + +For example: + +$$ + \frac{2}{8} + \frac{3}{8} = \frac{5}{8} +$$ + +The same applies to subtraction: + +$$ + \frac{a}{c} - \frac{b}{c} = \frac{a - b}{c} +$$ + + +## Adding/ subracting fractions without common denominators + +* Find the lowest common denominator for the two fractions +* Use this to create two equivalent fractions +* Add/subtract +* Reduce + +Demonstration: $\frac{4}{9} + \frac{1}{6}$ : + +The lowest common denominator is **the smallest number divisible by both of the denominators of the fractions without remainder**. diff --git a/Mathematics/Prealgebra/Dividing_fractions.md b/Mathematics/Prealgebra/Dividing_fractions.md index 459bf4a..5a12a45 100644 --- a/Mathematics/Prealgebra/Dividing_fractions.md +++ b/Mathematics/Prealgebra/Dividing_fractions.md @@ -6,23 +6,26 @@ tags: - division --- +# Dividing fractions + Suppose you have the following shape: -![draw.io-Page-9.drawio 1.png](../../img/draw.io-Page-9.drawio%201.png) +![draw.io-Page-9.drawio 1.png](../../img/draw.io-Page-9.drawio.png) One part is shaded. This represents one-eighth of the original shape. ![one-eighth-a.png](../../img/one-eighth-a.png) -Now imagine there are four instances of the shape and one-eighth remains shaded. How man one-eighths are there in four? +Now imagine there are four instances of the shape and one-eighth remains shaded. How man one-eighths are there in four? -![draw.io-Page-9.drawio 2.png](../../img/draw.io-Page-9.drawio%202.png) +![draw.io-Page-9.drawio 2.png](../../img/draw.io-Page-9.drawio.png) The shaded proportion represents $\frac{1}{8}$ of the shape. Imagine four of these shapes, how many eighths are there? This is a division statement: to find how many one-eighths there are we would calculate: + $$ -4 \div \frac{1}{8} +4 \div \frac{1}{8} $$ But actually it makes more sense to think of this as a multiplication. There are four shapes of eight parts meaning there are $4 \cdot 8$ parts in total, 32. One of these parts is shaded making it equal to $\frac{1}{32}$. @@ -33,37 +36,37 @@ $$ 4 \div \frac{1}{8} = 4 \cdot 8 = 32 $$ -Note that we omit the numerator but that technically the answer would be $\frac{1}{32}$. +Note that we omit the numerator but that technically the answer would be $\frac{1}{32}$. ### Formal specification of how to divide fractions We combine the foregoing (that it is easier to divide by fractional amounts using multiplication) with the concept of a [reciprocol](Reciprocals.md) to arrive at a definitive method for dividing two fractions. -It boils down to: *invert and multiply*: +It boils down to: _invert and multiply_: - > - > If $\frac{a}{b}$ and $\frac{c}{d}$ are fractions then: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$ +If $\frac{a}{b}$ and $\frac{c}{d}$ are fractions then: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$ -We invert the divisor (the second factor) and change the operator from division to multiplication. +We invert the divisor (the second factor) and change the operator from division to multiplication. #### Demonstration -*Divide $\frac{1}{2}$ by $\frac{3}{5}$* +Divide $\frac{1}{2}$ by $\frac{3}{5}$ $$ -\\begin{split} -\\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \cdot \frac{5}{3} \\ +\begin{split} +\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \cdot \frac{5}{3} \\ = \frac{5}{5} -\\end{split} +\end{split} $$ -*Divide $\frac{-6}{x}$ by $\frac{-12}{x^2}$* +Divide $\frac{-6}{x}$ by $\frac{-12}{x^2}$ $$ -\\begin{split} -\\frac{-6}{x} \div \frac{12}{x^2} = \frac{-6}{x} \cdot \frac{x^2}{-12} \\ = -\\frac{(\cancel{3} \cdot \cancel{2} )}{\cancel{x}} \cdot \frac{(\cancel{x} \cdot \cancel{x} )}{\cancel{3} \cdot \cancel{2} \cdot 2} \\ = -\\frac{x}{2} +\begin{split} +\frac{-6}{x} \div \frac{12}{x^2} = \frac{-6}{x} \cdot \frac{x^2}{-12} \\ = +\frac{(\cancel{3} \cdot \cancel{2} )}{\cancel{x}} \cdot \frac{(\cancel{x} \cdot \cancel{x} )}{\cancel{3} \cdot \cancel{2} \cdot 2} \\ = +\frac{x}{2} + +\end{split} -\\end{split} $$ diff --git a/Mathematics/Prealgebra/Handling_negative_fractions.md b/Mathematics/Prealgebra/Handling_negative_fractions.md index b642a61..4fc5fbe 100644 --- a/Mathematics/Prealgebra/Handling_negative_fractions.md +++ b/Mathematics/Prealgebra/Handling_negative_fractions.md @@ -6,31 +6,33 @@ tags: - negative-numbers --- +# Negative fractions + To work with negative fractions we draw on the [Rules for operations on like and unlike terms](Rules%20for%20operations%20on%20like%20and%20unlike%20terms.md). ## Fractions with unlike terms -* A fraction is just one number divided by another. $\frac{5}{10}$ is just ten divided by 5. +- A fraction is just one number divided by another. $\frac{5}{10}$ is just ten divided by 5. -* A positive integer divided by a negative or vice versa will always result in a negative. Thus $\frac{5}{-15}$ is equal to $-3$. +- A positive integer divided by a negative or vice versa will always result in a negative. Thus $\frac{5}{-15}$ is equal to $-3$. + +- We can therefore express the whole fraction as a negative: -* We can therefore express the whole fraction as a negative: $$ - \frac{5}{15} - $$ + $$ -* Or we could apply the negative symbol to the numerator. It would stand for the same value: +- Or we could apply the negative symbol to the numerator. It would stand for the same value: $$ \\frac{-5}{15} $$ Therefore: - > - > Let $a,b$ be any integers. The following three fractions are [equivalent](Equivalent%20fractions.md): $$\frac{-5}{15}, \frac{5}{-15}, - \frac{5}{15}$$ +> Let $a,b$ be any integers. The following three fractions are [equivalent](Equivalent%20fractions.md): $$\frac{-5}{15}, \frac{5}{-15}, - \frac{5}{15}$$ ## Fractions with like terms -* In cases where both the numerator and denominator are both negative, the value that the fraction represents will be positive overall. This is because the quotient of a negative integer divided by a negative integer will always be positive. +- In cases where both the numerator and denominator are both negative, the value that the fraction represents will be positive overall. This is because the quotient of a negative integer divided by a negative integer will always be positive. -* Thus: $$ \frac{- 12xy^2}{ - 18xy^2} = \frac{12xy^2}{18xy^2}$$ +- Thus: $$ \frac{- 12xy^2}{ - 18xy^2} = \frac{12xy^2}{18xy^2}$$ diff --git a/Mathematics/Prealgebra/Least_common_denominator.md b/Mathematics/Prealgebra/Least_common_denominator.md new file mode 100644 index 0000000..e69de29 diff --git a/Mathematics/Prealgebra/Reciprocals.md b/Mathematics/Prealgebra/Reciprocals.md index 6143b62..7b54c35 100644 --- a/Mathematics/Prealgebra/Reciprocals.md +++ b/Mathematics/Prealgebra/Reciprocals.md @@ -7,21 +7,22 @@ tags: - theorems-axioms-laws --- +# Recipricols + The [Property of Multiplicative Identity](Multiplicative%20identity.md) applies to fractions as well as to whole numbers: $$ -\\frac{a}{b} \cdot 1 = \frac{a}{b} +\frac{a}{b} \cdot 1 = \frac{a}{b} $$ With fractions there is a related property: the **Multiplicative Inverse**. - > - > If $\frac{a}{b}$ is any fraction, the fraction $\frac{b}{a}$ is called the *multiplicative inverse* or *reciprocol* of $\frac{a}{b}$. The product of a fraction multiplied by its reciprocol will always be 1. $$ \frac{a}{b} \cdot \frac{b}{a} = 1$$ +> If $\frac{a}{b}$ is any fraction, the fraction $\frac{b}{a}$ is called the _multiplicative inverse_ or _reciprocol_ of $\frac{a}{b}$. 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