Lattice points visible from the origin « The Lumber Room [A test of LaTeX-to-Wordpress conversion. Bugs remain, point them out. Original PDF] Here is a problem I love. It is simple to state, and it has a solution that is not trivial, but is easy to understand. [The solution is not mine. Question. Let us first imagine that we are standing at the origin, and that the grid is that of the lattice (integer) points. The blue points are visible; the grey points are not We now want to examine the question of which pairs are visible from the origin. not visible? is not visible because the point is "in the way", and the point obscures it. is not visible precisely when there is another lattice point blocking it, which is when there is an integer pair such that . is not visible precisely when there is an integer dividing both and , and is visible when there is no such integer, i.e. when have no common factor. For two "random" integers and , what is the probability that they have no common factor? For this to happen, it must be the case that no prime number . is . to .
What Is A Differential Equation? A differential equation can look pretty intimidating, with lots of fancy math symbols. But the idea behind it is actually fairly simple: A differential equation states how a rate of change (a "differential") in one variable is related to other variables. For example, the single spring simulation has two variables: time t and the amount of stretch in the spring, x. If we set x = 0 to be the position of the block when the spring is unstretched, then x represents both the position of the block and the stretch in the spring. the rate of change in velocity is proportional to the position For instance, when the position is zero (ie. the spring is neither stretched nor compressed) then the velocity is not changing. On the other hand, when the position is large (ie. the string is very much stretched or compressed) then the rate of change of the velocity is large, because the spring is exerting a lot of force. What is a Solution to a Differential Equation? Initial Conditions
Classifying Differential Equations When you study differential equations, it is kind of like botany. You learn to look at an equation and classify it into a certain group. The reason is that the techniques for solving differential equations are common to these various classification groups. And sometimes you can transform an equation of one type into an equivalent equation of another type, so that you can use easier solution techniques. First Order, Second Order, etc. The order of a differential equation is equal to the highest derivative in the equation. Linear vs. Linear just means that the variable in an equation appears only with a power of one. In math and physics, linear generally means "simple" and non-linear means "complicated". Recall that the equation for a line is y = m x + b where m, b are constants ( m is the slope, and b is the y-intercept). Homogeneous vs. This is another way of classifying differential equations.
Equation The first use of an equals sign, equivalent to 14x + 15 = 71 in modern notation. From The Whetstone of Witte by Robert Recorde (1557). The = symbol was invented by Robert Recorde (1510–1558), who considered that nothing could be more equal than parallel straight lines with the same length. Parameters and unknowns[edit] Equations often contain terms other than the unknowns. A system of equations is a set of simultaneous equations, usually in several unknowns, for which the common solutions are sought. has the unique solution x = −1, y = 1. Analogous illustration[edit] Illustration of a simple equation; x, y, z are real numbers, analogous to weights. A weighing scale, balance, or seesaw is often presented as an analogy to an equation. In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, and each of x, y, and z has a different weight. Types of equations[edit] Identities[edit] , which is true for all values of θ. use the identity:
Mathematical Equations Solving Linear Equations Math series Linear Equation: a mathematical expression that has an equal sign and linear expressions Variable:a number that you don't know, often represented by "x" or "y" but any letter will do! Variable(s) in linear expressions Cannot have exponents (or powers) For example, x squared or x2 Cannot multiply or divide each other For example: "x" times "y" or xy; "x" divided by "y" or x/y Cannot be found under a root sign or square root sign (sqrt) For example: √x or the "square root x"; sqrt (x) Linear Expression: a mathematical statement that performs functions of addition, subtraction, multiplication, and division These are examples of linear expressions: These are not linear expressions: Solve these linear equations by clicking and dragging a number to the "other" side of the equal sign. More examples: Linear equation, solving example #1: Find x if: 2x + 4 = 10 Linear equation, solving example #2: Find x if: 3x - 4 = -10(using negatives) Linear equation, solving example #3: Math series