A varied diet generally provides enough of each vitamin and mineral. However, some people may need supplements to correct deficiencies of particular vitamins or minerals.
This is a correct statement.
All animals including people, birds, dogs, cats, fish and reptiles should get sufficient nutrients if they get a varied diet. However, if they have digestive problems (irritable bowel, coeliacs, or other issues that inhibit the absorption of nutrients from food), or they are under stress, have a virus, suffer from lack of sleep, have ulcers or have a major health issue like cancer, or they are not getting good quality food or food on a regular basis, then a vitamin and or mineral supplement taken several times a day, should help improve their overall health.
Having said this, the vitamins/ minerals also need to be in a form that is easily digested/ absorbed by the animal. Lots of cheap brands of vitamin/ mineral supplements use oxides (zinc oxide, magnesium oxide, etc) and most animals cannot digest oxides, and these can actually irritate the intestine and bowel and make digestive issues worse.
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Most aquarium fish do not get a varied diet and most commercially prepared dry fish foods absorb moisture from the air very quickly, and moisture destroys vitamins. If you have a container of dry fish food that has been open for more than 1 month, the vitamins in it have probably degraded and will be of little or no value.
The best way to keep dry foods is in an air tight container with a moisture absorbing gell pack. Alternatively, when you get a new container of dry food, put most of it into plastic ziplock bags, squeeze all the air out and seal it up. Put that in another ziplock bag and freeze it or put it in a cool dry area away from moisture or humidity. Keep just enough food in a small container to use in 1 week. After a week, take a bit of the food that has been kept in ziplock bags and add that to the food container. Basically keep most of the food away from air and moisture and only have 1 weeks worth of food open at any time.
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Fish should not need vitamin A supplements because most fish food is made from fish meal, and fish meat and their organs contains a lot of vitamin A. So they should get sufficient vitamin A from fish food.
Betacarotene is readily absorbed by most animals including fish and can be stored in the body for long periods of time. You cannot really overdose on natural sources of Betacarotene (carrot, squash, pumpkin, apricots, etc) but excessive intake can discolour the skin and make it orange. However, you require very high doses for a long period of time for this to happen, and there are no major health implications associated with this besides looking like an orange.
One of the advantages of Betacarotene is the body can convert it into vitamin A when required. Animals fed lots of Betacarotene will not need vitamin A supplements because they can produce their own vitamin A whenever they need it, from the Betacarotene.
Vitamin A is poisonous in high doses and Betacarotene from natural sources isn't. This makes Betacarotene a much safer choice for all animals and fish and allows each individual organism to make its own vitamin A when they require it.
Fish and birds regularly fed Betacarotene supplements show more orange and red colouration compared to the same animals that are not fed it.
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Vitamin C is readily available in any plant matter and blending or cutting up soft leaved aquarium plants and feeding them to the fish will provide vitamin C. If you don't want to use aquarium plants you can use algae or dark green leafy vegetables like spinach and silverbeet. However, aquatic plants & algae are more readily absorbed by animals and should be the first choice.
Vitamin B & E are also found in algae, as are a number of minerals. This makes algae a useful and nutritious food for people and fish.
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Vitamin D is made when skin is exposed to ultra violet light (found in sunlight). This is an interesting subject because fish from deep water don't get any ultra violet (uv) light, whereas surface dwelling fish get lots. Plankton (baby fish, shrimp, algae, etc) in the rivers and ocean live within the top 12 inches of the water and get heaps of uv light but as the fish grow and move into deeper water they get less uv light. Fish found in blackwater (stained with tannins from plants) or in steams that have lots of trees shading the water are not exposed to high levels of uv light. And fish that hide in plants or caves don't get as much uv light.
You can get uv light globes from shops but they can damage your eyes and cause sunburn to people sitting near the aquarium and generally are not recommended. However, corals and some marine fishes have shown improved colouration when exposed to uv light in aquariums.
As a general rule, most aquarium fish do not need a uv light above their aquarium but further study into this area might suggest otherwise. Perhaps fish do require some uv light to keep their vitamin D levels up, or perhaps they don't need as much vitamin D as terrestrial animals, which is the most likely case due to water filtering out the light.
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As a general rule, if a fish or animal is healthy and fed a varied diet that it can easily digest, there should be no need for vitamin supplements. However, there have been lots of studies that show vitamin supplements given to animals fed a nutritious and varied diet, actually live longer and have healthier lives compared to animals fed the same diets without the supplements.
If the fish is old or unwell, then daily supplements might help, assuming other factors like disease and water quality/ chemistry are not causing the issue. In the OP's case, supplements might help, but choose the right type because fish and humans have different requirements, and fish are tiny compared to people.
If you can't get a fish supplement, then a bird or reptile supplement is safer than a human supplement. Or just mix algae with raw finely chopped prawn.
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In my experience, most male Bettas only live for a couple of years and whilst there might be exceptions with some living 5 years or more, there is no way of telling what the fish has been exposed to prior to the OP obtaining the fish. It might have been exposed to chemicals, poisoned, malnourished or been kept in poor conditions. Inbreeding and genetic deformities like long fins, short bodies, etc would also contribute to a shorter life span.