Na element group12/30/2023 Sodium is the fourth most abundant element on earth, comprising about 2.6% of the earth's crust. Most likely the second most useful compound is sodium carbonate, also known as washing soda, which is used as a water softener.īaking soda also contains sodium and is used in cooking. The most common sodium compound is sodium chloride, which is normal table salt, used on food and to throw on iced roads. Sodium is also used as a reagent in the chemical industry.īut the uses for sodium compounds are much greater than that of pure sodium metal. Liquid sodium is used as a heat exchanger in nuclear power plants. Pure sodium is used in the production of titanium. Similar to lithium, sodium is usually stored in a closed container in kerosene or mineral oil, to prevent or limit the reaction with oxygen in the air. Sodium is a soft silver-white metal, with a similar appearance to lithium. Pure sodium will cause a violent exothermal reaction when it comes into contact with water. But the element behind these compounds is much less innocent. Sodium chloride and sodium bicarbonate are compounds used daily in every household. When we have a*x = b, where a and b were in a group G, the properties of a group tell us that there is one solution for x, and that this solution is also in G.Sodium is the wonderful metal in common table salt, but did you know it floats on water? Here is one good reason: Solving EquationsĪs it turns out, the special properties of Groups have everything to do with solving equations. They are even used to tell if polynomials have solutions we can find. They are used by space probes so that if data is misread, it can be corrected. Not because there isn't a good one, but because the applications of groups are very advanced.įor example, they are used on your credit cards to make sure the numbers scanned are correct. So why do we care about these groups? Well, that's a hard question to answer. So we have shown that using one operation, the integers are a group, and under another, they aren't. So what is 5 -1? It's 1/5.īut that isn't in the integers! Ahhhh! The integers don't contain multiplicative inverses, so they can't be a group with respect to multiplication. So if we take a number a, can we find a -1 such that a * a -1 = e? Let's try 5 again. Now we need to find out if integers under multiplication have inverses. So it is closed under the operation.Īnd we're done! is a group under multiplication. But it should be pretty obvious that it is.įinally, is it closed? Is 1*1 in the group? Yep. Well, since we have only 2 numbers, we can try every possibility. Is it associative? a * (b * c) = (a * b) * c. So if a = -1, then a -1 = -1 as well! Since we have found an inverse for every element, we know the group is closed with respect to inverses. So let's start off with 1.ġ * 1 = 1, so we know that if a = 1, a -1 = 1 as well. If we have a in the group, then we need to be able to find an a -1 such that a * a -1 = 1 (or rather, e). First, is there an identity? Well, this is going to be easy, there are only three possibilities. If a word is defined well, you know exactly what I mean when I say it.īack to the four steps. Think about applying those two words, "defined well" to the English language. One thing about operators is that they must be well defined. You will learn in a minute that there are really only two! Well Defined Now above it looks like there are 3 operations. (Also note: division is not included, because it also returns a remainder) It still takes two elements, even if they are the exact same elements. These all take two numbers and combine them in different ways to get one number. You already know a few binary operators, even though you may not know that you know them: A binary operation is just like an operation, except that it takes 2 elements, no more, no less, and combines them into one. So we will now be a little bit more specific. So far we have been a little bit too general. So "red mixed with blue makes purple" becomes "red + blue = purple". So I'm going to let "mixed with" be symbolized by + and "makes" be symbolized by =. If I have to write a lot, I'm going to want to shorten that up. So for example, red mixed with green makes yellow, and red mixed with blue makes purple.īut saying "red mixed with blue makes purple" is long and annoying.
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