The Difference Of Squares

[Timothee Cazares]

Inmathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity

Since the two factors found by this method are complex conjugates, we can use this in reverse as a method of multiplying a complex number to get a real number. This is used to get real denominators in complex fractions.[1]

The difference of two squares can also be used in the rationalising of irrational denominators.[2] This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots.

The difference of two squares can also be used as an arithmetical short cut. If two numbers (whose average is a number which is easily squared) are multiplied, the difference of two squares can be used to give you the product of the original two numbers.

A ramification of the difference of consecutive squares, Galileo's law of odd numbers states that the distance covered by an object falling without resistance in uniform gravity in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain distance during an arbitrary time interval, it will cover 3, 5, 7, etc. times that distance in the subsequent time intervals of the same length.

The proof is identical. For the special case that a and b have equal norms (which means that their dot squares are equal), this demonstrates analytically the fact that two diagonals of a rhombus are perpendicular. This follows from the left side of the equation being equal to zero, requiring the right side to equal zero as well, and so the vector sum of a + b (the long diagonal of the rhombus) dotted with the vector difference a - b (the short diagonal of the rhombus) must equal zero, which indicates the diagonals are perpendicular.

However, if you need to do the computation manually, it's best to group arguments with similar magnitudes. This means the second option is more precise, especially when cos_theta is close to 0, where precision matters the most.

In other words, taking x - y, x + y, and the product (x - y)(x + y) each introduce rounding errors (3 steps of rounding error). x2, y2, and the subtraction x2 - y2 also each introduce rounding errors, but the rounding error obtained by squaring a relatively small number (the smaller of x and y) is so negligible that there are effectively only two steps of rounding error, making the difference of squares more precise.

As an aside, you will always have a problem when theta is small, because the cosine is flat around theta = 0. If theta is between -0.0001 and 0.0001 then cos(theta) in float is exactly one, so your sin_theta will be exactly zero.

To answer your question, when cos_theta is close to one (corresponding to a small theta), your second computation is clearly more accurate. This is shown by the following program, that lists the absolute and relative errors for both computations for various values of cos_theta. The errors are computed by comparing against a value which is computed with 200 bits of precision, using GNU MP library, and then converted to a float.

[Edited for major think-o] It looks to me like option 2 will be better, because for a number like 0.000001 for example option 1 will return the sine as 1 while option will return a number just smaller than 1.

No difference in my option since (1-x) preserves the precision not effecting the carried bit. Then for (1+x) the same is true. Then the only thing effecting the carry bit precision is the multiplication. So in both cases there is one single multiplication, so they are both as likely to give the same carry bit error.

This is a factoring calculator if specifically for the factorization of the difference of two squares. If the input equation can be put in the form of a2 - b2 it will be factored. The work for the solution will be shown for factoring out any greatest common factors then calculating a difference of 2 squares using the idenity:

If a is negative and we have addition such that we have -a2 + b2 the equation can be rearranged to the form of b2 - a2which is the correct equation only the letters a and b are switched; we can just rename our terms.

Whenever you have a binomial with each term being squared (having an exponent of [latex]2[/latex]), and they have subtraction as the middle sign, you are guaranteed to have the case of difference of two squares.

These are other ways to write the formula of the difference of two squares using variables. Learn to recognize them in various appearances so that you know exactly how to handle them.

The first term of the binomial is definitely a perfect square because the variable [latex]x[/latex] is being raised to the second power. However, the second term of the binomial is not written as a square. So we need to rewrite it in such a way that [latex]9[/latex] is expressed as some number with a power of [latex]2[/latex]. I hope you can see that [latex]9 = \left( 3 \right)^2[/latex]. Clearly, we have a difference of two squares because the sign between the two squared terms is subtraction.

For this example, the solution is broken down in just a few steps to highlight the procedure. Once you get comfortable with the process, you can skip a lot of steps. In fact, you can go straight from the difference of two squares to its factors.

Now, we can truly rewrite this binomial as the difference of two squares with distinct terms that are being raised to the second power; where [latex]16y^4 = \left( 4y^2 \right)^2[/latex] and [latex]81 = \left( 9 \right)^2[/latex]

Are we done already? Well, examine carefully the binomials you factored out. The second parenthesis is possibly a case of difference of two squares as well since [latex]4y^2 = \left( 2y \right)\left( 2y \right)[/latex] and clearly, [latex]9 = \left( 3 \right)\left( 3 \right)[/latex].

Now we can deal with the binomial inside the parenthesis. It is actually a difference of two squares because we can express each term of the binomial as an expression with a power of [latex]2[/latex].

You may keep it in that form as your final answer. But the best answer is to combine like terms by adding or subtracting the constants. This also simplifies the answer by getting rid of the inner parenthesis.

Not sure if the binomial you've factoring is a difference of squares problem? This tutorial will show you what characteristics the binomial must have in order to be a difference of squares problem. Take a look!

Anytime you square an integer, the result is a perfect square! The numbers 4, 9, 16, and 25 are just a few perfect squares, but there are infinitely more! Check out this tutorial, and then see if you can find some more perfect squares!

Monomials are just math expressions with a bunch of numbers and variables multiplied together, and one way to compare monomials is to keep track of the degree. So what's a degree? Well, if you've ever wondered what 'degree' means, then this is the tutorial for you.

There is a bunch of vocabulary that you just need to know when it comes to algebra, and coefficient is one of the key words that you have to feel 100% comfortable with. Check out the tutorial and let us know if you want to learn more about coefficients!

Polynomials are those expressions that have variables raised to all sorts of powers and multiplied by all types of numbers. When you work with polynomials you need to know a bit of vocabulary, and one of the words you need to feel comfortable with is 'term'. So check out this tutorial, where you'll learn exactly what a 'term' in a polynomial is all about.

I am finding it difficult to adjust the end positions of horizontal and vertical lines that I am drawing. I am constructing tree diagrams and I find that I always need to keep moving them around and making them longer or shorter.

I select a line and then try and drag one of the blue squares that appear at the end points of the line to the desired position. This is not easy. The point often appears not to move and when I release the mouse button, the point jumps to the wrong place, or backwards, or even to the page margin.

From my perspective, the larger square indicates the starting point and the smaller square indicates the end point of the line - just create a line from left to right (larger square is left) or from right to left (larger square is right). This is important to find which Arrow Styles (called Start style and End style) to change, if you need to add arrows to the start or to the end of the line.

To align correctly several connectors (to give the impression that all terminal segments come from a single wide horizontal line), take care that all all destination boxes have the same height (use F4 to access the properties and be able to force exact measures) and are all aligned so that their top border have the same coordinate (this is more important than the height itself). Adapt if your chart follows a perpendicular direction.

To celebrate my birthday, I like to find interesting number theoreticproperties of my new age. My upcoming 61st birthday was challenging, but thenI noticed that $61 = 5^2 + 6^2 = 5^3 - 4^3$, the sum of two consecutive squaresand the difference of two consecutive cubes. I wondered what other numbershad this property; that is, the integer solutions to$a^2+(a+1)^2 = (b+1)^3 - b^3$ or equivalently $2a^2+2a = 3b^2+3b$.I ran an experiment in Matlab and got the following striking results.

To parametrise all rational solutions of $2a(a+1)=3b(b+1)$ you can now homogenise $$2(2a+1)^2+1=3(2b+1)^2$$into $$2(2a+1)^2+c^2=3(2b+1)^2$$ and define $$x:=2a+1,y:=c,z:=2b+1,$$so that $$2(2a+1)^2+c^2=3(2b+1)^2$$ becomes $$2x^2+y^2-3z^2=0.$$This is a diagonal conic with the rational point $(x_0,y_0,z_0)=(1,1,1)$ and you can find the parametrisation of all solutions before Theorem 2.3 these notes.

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