The overall best solution ...

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The overall best solution might be to not Remove Calcium From Water but instead to treat the calcium in your water by installing a  Hard Water Bullet along with an in-line filter. If you cannot ingest calcium for health reasons, then you should install a high quality Reverse Osmosis Drinking Water System which will remove the calcium. The reason I say this is simple: You will not need to purchase salt, and you will not waste water because the Hard Water Bullet will not need to regenerate and you’re not drinking water with sodium chloride in it.

In a watery solution calcium is mainly present as Ca2+ (aq), but it may also occur as CaOH+ (aq) or Ca(OH)2 (aq), or as CaSO4 in seawater.
Calcium is an important determinant of water harness, and it also functions as a pH stabilizer, because of its buffering qualities. Calcium also gives water a better taste.

In what way and in what form does calcium react with water?

Contrary to magnesium placed directly above calcium in the periodic chart, elementary calcium reacts with water at room temperature, according to the following reaction mechanism:

Ca (s) + 2H2O (g) -> Ca(OH)2 (aq) + H2 (g)

This reaction forms calcium hydroxide that dissolves in water as a soda, and hydrogen gas.

Other important calcium reaction mechanisms are erosion reactions. These usually occur when carbon dioxide is present. Under normal conditions calcium carbonate is water insoluble. When carbon dioxide is present carbonic acid is formed, affecting calcium compounds.

The reaction mechanism for carbon weathering is:

H2O + CO2 -> H2CO3  and CaCO3 + H2CO3 -> Ca(HCO3)2

And the total reaction mechanism:

CaCO3 (s) + CO2 (g) + 2H2O (l) -> Ca2+ (aq) + 2 HCO3- (aq)

The product is calcium hydrogen carbonate.

Solubility of calcium and calcium compounds

Elementary calcium reacts with water. Calcium compounds are more or less water soluble. Calcium carbonate has a solubility of 14 mg/L, which is multiplied by a factor five in presence of carbon dioxide. Calcium phosphate solubility is 20 mg/L, and that of calcium fluoride is 16 mg/L. Calcium chromate solubility is 170 g/L, and at 0oC calcium hypo chlorate solubility is 218 g/L. Solubility of other calcium compounds lies between the levels of these examples, for example calcium arsenate 140 mg/L, calcium hydroxide 1.3 g/L and calcium sulphate 2.7-8.8 g/L.