topic_notes/maths_revision.html

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    <h1>Maths Revision</h1>
    <center>Notes on key GCSE Maths topics I've had to revise in order to learn the necessary statistics for Data Science. </center>

    <br>
    <a href="/topics.html"> <<< go back to Topics </a>
    <br>
    
    
    <h2>Fractions</h2>  

    Numerator = number on top 
    <br>
    Denominator = number on bottom 

    <h3>Adding Fractions</h3>

    Convert denominators (multiply top and bottom by same number) so they're the same.
    <br>
    Add the numerators and keep the denominator the same.
    <br><br>

    1/3 + 1/6 
    <br>
    = 2/6 + 1/6
    <br>
    = 3/6
   
    <br>

    <h3>Subtracting Fractions</h3>

    Same as above but subtract instead of add.

    <br>

    <h3>Multiplying Fractions</h3>
    Multiple top numbers, and multiply bottom numbers.
    <br><br>
    1/5 x 2/3
    <br>
    = (1x2)/(5x3)
    <br>
    = 2/15

    <br>

    <h3>Dividing Fractions</h3>
    Flip the 2nd fraction and multiply.
    <br><br>
    (1/2) / (1/6)
    <br>
    = (1/2) x (6/1)
    <br>
    = 6/2
    <br>
    = 3 

    <br><br>

    <h2>Standard Form</h2>  

    4.3 x 10^4
    <br>
    = 43000.0
    <br>
    Move the decimal place to the right
    <br><br>

    4.3 x 10^-4
    <br>
    = 0.00043
    <br>
    Move the decimal place to the left.
    <br><br>

    <h2>Surds</h2>  

    SQRT(5)^2
    <br>
    = 5
    <br><br>

    2x(SQRT(5)) + 3x(SQRT(5))
    <br>
    = 5x(SQRT(5))
    <br><br>

    2x(SQRT(5)) x 3x(SQRT(5))
    <br>
    = SQRT(2^2 x 5) x SQRT(3^2 x 5)
    <br>
    = SQRT(4 x 5) x SQRT(9 x 5)
    <br>
    = SQRT(20) x SQRT(45)
    <br>
    = SQRT(20 x 25)
    <br>
    = SQRT(900)
    <br>
    = SQRT(100 x 9)
    <br>
    = SQRT(100) x SQRT(9)
    <br>
    = 10 x 3
    <br>
    = 30
    <br><br>

    <h2>Fractional Indices</h2>  

    5^2 = 25
    <br><br>
    
    25^(1/2) = SQRT(25) = 5
    <br><br>
    
    2^3 = 2 x 2 x 2 = 8
    <br><br>
    
    8^(1/3) = CUBEROOT(8) = 2
    <br><br>
    
    3^(1/2) x 3^(1/2) = 3^1 = 3
    <br><br>
    
    25^(3/2) = (25^(1/2))^3 = SQRT(25)^3 = 5^3 = 5 x 5 x 5 = 125
    <br><br>

    <h2>Algebra</h2> 

    2(y + 4) = 2y + 8
    <br><br>
    
    3y + 12 = 3(y + 4)
    <br><br>
    
    (y + 3 )(y + 4) 
    <br>
    = y^2 + 4y + 3y + 12 
    <br>
    = y^2 + 7y + 12
    <br><br>
    
    y^2 + 6y + 8
    <br>
    = (y + 2)(y + 4)
    <br><br>
    
    y^3 x y^7 = y^10 (multipling terms with indices involves adding thr indices together)
    <br><br>
    
    y^6 / y^2 = y^4 (dividing terms with indices involves 
    subtracting their indices)
    <br><br>
    
    y = ax + c (a is the gradient, c is the intercept on the y axis)
    <br><br>

     <h2>Quadratic Equations</h2> 

     y^2 + 3y - 4 = 0
     <br>
     => (y + 4)(y - 1) = 0
     <br>
     => y + 4 = 0 and y - 1 = 0
     <br>
     => y = -4, and y = 1
     <br><br>
     
     x^2 + 6x + 7
     <br>
     = (x + 3)^2 - 2
     <br>
     ...then you can re-arrange this to find the value of x (and then the value of y) where the quadratic line on the plot has zero gradient (turning point) - this is useful in machine learning.
     <br>
     e.g. (x + 3)^2 - 2 = 0
     <br>
     => (x + 3)^2 = 2
     <br>
     => x + 3 = SQRT(2)
     <br>
     => x = SQRT(2) - 3
     <br><br>

     <h2>Trigonometry</h2>

     To find the length of a side of a right angled triangle, where c is the longest side:
     <br>
     a^2 + b^2 = c^2
     <br>
     This is useful for fiding the distance between points on a graph which is useful in machine learning.
     <br><br>

     The area of a triangle is:
     <br>
     (width x height) / 2
     <br>
     This is useful in statistics (and therefore machine learning) when trying to find a probability via calculating the area under a probability distribution graph.
     <br><br>


     <h2>Vectors</h2>

     (1, 2) + (3, 4) = (4, 6)
     <br>
     (1, 2) - (3, 4) = (-2, -2)
     <br>
     2 x (1, 2) = (2, 4)
     <br>
     (2, 4) / 2 = (1, 2)
     <br><br>


     <h2>Factorials</h2>

     n! = n(n-1)(n-2)...(1)
     <br>
     4! = 4 x 3 x 2 x 1
     <br>
     5! = 5 x 4 x 3 x 2 x 1
     <br>
     0! = 1
     <br>
     1! = 1
     <br><br>

     The number of ways to arrnage n items is n!
     <br>
     e.g. the number of ways to arrange 3 items is 3! = 3 x 2 x 1 = 6.
     <br><br><br>


  Useful video: <a href="https://www.youtube.com/watch?v=dYPwEyeyPLA">"The whole of GCSE 9-1 Maths in only 2 hours!!" by Science and Maths by Primrose Kitten</a>
    <br>
    <br>
    
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