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Mathematics - Coggle Diagram

- - - - Real numbers are rational or irrational, where rational numbers can be expressed as a fraction
    - - A prime number is a number that can only be divided by 1 and itself, where prime factorisation is the process of expressing a composite number as the product of its prime factors
    - - HCF is the largest factor common to all numbers. LCM is the smallest multiple common to all numbers
    - - A perfect square is a number whose square root is a whole number
    - - A perfect cube is a number whose cube root is a whole number
    - - The reciprocal of x over y is y over x
    - - All non-0 digits are significant, though 0s between non-0 digits are significant. In decimals, 0s only after a non-0 digit are significant
    - - Replace all no. with aproximate values
    - - Written A x 10^n, where A is smaller than equals to 1 and smaller than 10 and n is an integer
      - 10^12 trillion tera- T
        10^9 billion giga- G
        10^6 million mega- M
        10^3 thousand kilo- k
        10^-3 thousandth milli- m
        10^-6 millionth micro- μ
        10^-9 billionth nano- n
        10^-12 trillionth pico- p
    - - When 2 identical bases with differing indices m and n are multiplied, simply take base to the power of m+n
        When 2 identical bases with differing indices m and n are divided, simply take the bae to the power of m-n
        (a^m)^n = a^mn
        a^n b^n = (ab)^n
        a^n / b^n = (a/b)^n
  - - - a/b = a:b
    - - Where ratio is 1:r, that means 1 unit on the map corresponds to r units in real life. The map scale it is 1:r^2
    - - y=kx
    - - y= k/x
  - - - x%= x/100
    - - Percentage change = change/original x100
  - - - Measures how one quantity changes with another quantity
    - - 1m=100cm
        1km=1000cm 1
        h=60min
        1min=60s
        1 m/s = 3.6 km/h
    - - Speed= Distance/Time
      - Average Speed= Total distance / Total time
  - - - Terms compose a variable - number and a coefficient - unknown. Constants are numbers
    - - A formula is an eqaution used to find a value
    - - First number + common difference(n-1)
    - - (a+b)(c+d)=ac+ad+bc+bd
    - - (a±b)^2 = a^2 ± 2ab + b^2 and (a+b)(a-b) = a^2 - b^2
    - - Extract common factors or grp
    - - When adding or subtracting, the denominator must be common. When multiplying, simplify where possible. Take the multiplication of reciprocal for division
    - - Ensure the subject is isolated on a single side of equation
  - - - Graidient = rise/run and y=mx+c
      - Determine if a graph slopes up or down, find c, then substitute y=0 to find the x-intercept if needed
    - - y=ax^2 where a is positive, graph lies above x-axis. If a is negative, graph lies below x-axis
      - y=a(x-p)(x-q) where a is positive, minimum point is negative. Where a is negatuve, maximum point is positive
        Turning point is half of the sum of p and q
      - y = a(x-h)^2 + k where a is positive, graph curves downwards. Where a is negative, graph curves upwards
        Line of symmetry is x=h, and turning points (h,k)
    - - y=ax^3 where a is positive, graph appears in first and third quadrants. Where a is negative, graph appears in second and fourth quadrants
    - - y= a/x where a is positive, graph appears in first and third quadrants. Where a is negative, graph appears in second and fourth quadrants
      - y= a/ x^2 where a is positive, graph appears in first two quadrants. Where a is negative, graph appears in last two quadrants
    - - y= ka^x
        
        Where a and k are positive, graph curves from left to right upwards, and c=k
        
        Where a is below one yet positive and k is positive, graph curves donwards and c=k
        
        Where a is positive but k is negative, graph curves from left to right downwards, where c=k
        
        Where a is smaller than 1 yet positive and k is negative, graph curves from left ro right upwards, where c=k
    - - Draw a tangent and find m via y2-y1 / x2-x1
      - To solve simultaneous equation using graphical method, draw both and find intersection
  - - - Solve unknown
    - - Eliminatation - the absolute value of coefficient the same, then add or subtract the equations, forming a single linear one with only 1 unknwon
        Subsitution - make the varibale the subject of one equation then substitute it to solve another equation
        Graphical - draw graphs and find the intersection