+---------------------------------------------------------+
+---------------------------------------------------------+
| a repository of curiosity - [Home] [Software] [Books&Movies] [Web Security] [Binary Security] [Maths] [About] |
Maths bits and bobs I find interesting.
On Mathematical Maturity - Thomas Garrity
+---------------------------------------------------------+
+---------------------------------------------------------+
y = mx + b
the slope formula.
- the formula (m * x + b ) it contains a few components that have different roles, that when evaluated will equal a single value that we can map as a point on the cartisian plane. Doing a second evaluation of this expression will result in a second point,
providing with a line (resolution 2 points) and slope.
* (m) being a assigned static value that will determine the steepness of the slope. This coefficient determines the step multiple.
* (x) being a variable we can assign will result in a valid value on the slope line. By rotating through numbers and taking the point at each, we can build valid points for the equation, building out the slope.
* (b) being an offset that will statically shift the graph of the slope by a set amount. We can think of b = 0 as the natural slope position but we can keep the slope the same steepness but just alter the location on the y axis.
- Because the numbers in the expression are a multiplication and an addition, by statically setting an m and b, rotating values into the x position, we can build a slope that is linear.
- This concept in another context
We can think of this linear equation as moving a set amount each n times, can we increment n by one each step (n+1).
This is the general formula, providing we set b = 0.
We can look at this for the generalised formula for the times tables.
y = 4 * 1 + 0, y = 4 * 2 + 0, y = 4 * 3 + 0
<-------------------------------------------------->
x = |1| |2| |3|
<-------------------------------------------------->
log space
My experience learning logs was the application of logs was buried in the material not put up front and associated with the core principle. * The exponential function is a growth function. For every (x) step we increase by some non-linear amount. * The log function undoes this or is the inverse of the exponential function. * We are talking about rates of change that are not linear. Basic Logs ----------- log2(8) = 3 How many times do we need to multiply 2 in order to get the result 8. 2 * 2 * 2 - the answer is 3 - Okay cool trick, seem more like a number game than anything. How to mentally map logs --------------------------- - The application: We can normalise the values distributed across a graph. - Say we want to represent values that widely spread apart on the number line. This presents a problem when trying to visualise these huge gaps on a graph. Representing the large gaps and make data points with a small gap hard to see. - The normal base 10 magnitude for group of numbers scales from small gaps to very large gaps. tens, hundreds, thousands 10^1, 100^1, 1000^1 - With logs we can represent large distances between points on the number line, with small representation steps on our graph - In the base10 case multiplication is used creating large distances, addition in the log case creates a smaller scale to compress large ranges. - This makes data points on the graph with small deltas and data points with huge number deltas easy to read, as you can think of this as all numbers being normalised within a representation band for the graphoperator operands operations
Modes of operations (addition / multiplication)
- We can reduce the mathematical operations (addition, subtraction, multiplication, division) down to two. addition and multiplication.
Addition:
* Generally addition is thought of as always moving right on the number line. We start at our initial point, any addition must move us to a greater number to the right of our current location.
* Generally addition is thought of subtraction as always moving left on the number line. We start at our initial point, any subtraction must move us to a lower number to the left of our current location.
- The general form of addition is - (operand) Operator (operand)
- With addition we can move to the left in a negative direction by introducing negative numbers to either or both of our operands.
- The result is subtraction
Multiplication:
* The same concepts as before, generally we can think of multiiplication as an increase by a scale of x.
* By introducing numbers < 1 - in the form of fractions or decimals, our multiplication becomes division