Proving the Density of a Specific Set in Real Numbers

In summary, density is a measure of how much mass is contained in a given volume of a substance, and is commonly expressed in units of mass per unit volume. It is calculated by dividing the mass of an object by its volume, and the SI unit for density is kilograms per cubic meter. Density can affect the properties of a substance, such as buoyancy and conductivity, and is affected by temperature as substances tend to expand when heated and contract when cooled. In most cases, substances become less dense as they are heated and more dense as they are cooled.
  • #1
D.K.
12
0
Is there an easy way to prove that for any irrational [tex]\xi[/tex] the set:

[tex]\{x \in \mathbb{R}: x = p + q\xi, \ p, q \in \mathbb{Z}\}[/tex] is dense in [tex]\mathbb{R}[/tex]?

I know a proof involving notions from measure theory of which I unfortunately know nothing about. Any help would be very appreciated.
 
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  • #2
Nevermind, it turned out to be rather easy.
 

1. What is density?

Density is a measure of how much mass is contained in a given volume of a substance. It is commonly expressed in units of mass per unit volume, such as grams per cubic centimeter or kilograms per liter.

2. How is density calculated?

Density is calculated by dividing the mass of an object by its volume. The formula for density is: density = mass/volume.

3. What is the SI unit for density?

The SI unit for density is kilograms per cubic meter (kg/m^3). However, other common units for density include grams per cubic centimeter (g/cm^3) and pounds per cubic inch (lb/in^3).

4. How does density affect the properties of a substance?

Density can affect the properties of a substance in many ways. For example, substances with higher densities tend to sink in liquids with lower densities, and vice versa. Density can also affect the buoyancy, conductivity, and strength of a substance.

5. What is the relationship between density and temperature?

Density is affected by temperature, as substances tend to expand when heated and contract when cooled. This means that the density of a substance may change with temperature. In most cases, substances become less dense as they are heated and more dense as they are cooled.

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