# ASCIIMathML

The **Erdős–Anning theorem** states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. It is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945.^{[1]}

## Rationality versus integrality

Although there can be no infinite non-collinear set of points with integer distances, there are infinite non-collinear sets of points whose distances are rational numbers. For instance, on the unit circle, let *S* be the set of points for which is a rational number. For each such point, both and are themselves both rational, and if and define two points in *S*, then their distance is the rational number . More generally, a circle with radius contains a dense set of points at rational distances to each other if and only if is rational.^{[2]}

For any finite set *S* of points at rational distances from each other, it is possible to find a similar set of points at integer distances from each other, by expanding *S* by a factor of the least common denominator of the distances in *S*. Therefore, there exist arbitrarily large finite sets of points with integer distances from each other. However, including more points into *S* may cause the expansion factor to increase, so this construction does not allow infinite sets of points at rational distances to be translated to infinite sets of points at integer distances.

It remains unknown whether there exists a set of points at rational distances from each other that forms a dense subset of the Euclidean plane.^{[2]}

## Proof

To prove the Erdős–Anning theorem, it is helpful to state it more strongly, by providing a concrete bound on the number of points in a set with integer distances as a function of the maximum distance between the points. More specifically, if a set of three or more non-collinear points have integer distances, all at most some number , then at most points at integer distances can be added to the set.

To see this, let *A*, *B* and *C* be three non-collinear members of a set *S* of points with integer distances, all at most , and let , , and be the three distances between these three points. Let *X* be any other member of *S*. From the triangle inequality it follows that is a non-negative integer and is at most . For each of the integer values *i* in this range, the locus of points satisfying the equation forms a hyperbola with *A* and *B* as its foci, and *X* must lie on one of these hyperbolae. By a symmetric argument, *X* must also lie on on one of a family of hyperbolae having *B* and *C* as foci. Each pair of distinct hyperbolae, one defined by *A* and *B* and the second defined by *B* and *C*, can intersect in at most four points,
and every point of *S* (including *A*, *B*, and *C*) lies on one of these intersection points. There are at most intersection points of pairs of hyperbolae, and therefore at most points in *S*.

## Maximal point sets with integral distances

An alternative way of stating the theorem is that a non-collinear set of points in the plane with integer distances can only be extended by adding finitely many additional points, before no more points can be added. A set of points with both integer coordinates and integer distances, to which no more can be added while preserving both properties, forms an Erdős–Diophantine graph.

## References

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## External links

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^{2.0}^{2.1}Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.