This is how counting leads to the idea of division and even of division with remainder. In this case, addition and multiplication combine in the mathematical notation: 2♳ + 1 = 7. For example, splitting 7 into groups of 3 objects, we get 2 groups and 1 leftover object. Sometimes, we can't split objects evenly into smaller groups. 3♲ is thus the same as "1 and 1 and 1 and 1 and 1 and 1" which is denoted as 6: 3♲ = 6. For example, the mathematical notation for "2 and 2 and 2" is "3 times 2" or, symbolically, 3♲. If all the groups have the same amount of objects, we will discern the relevance of another arithmetic operation - multiplication - to counting. It is much easier to count objects by grouping them into small groups and later adding up the quantities associated with every group. While typing the quote I also lost the count several times. "She can't do addition," said the Red Queen.īut repeatedly adding 1 is a very primitive form of addition which is also not very useful. "I don't know, " said Alice, "I lost count." "What's one and one and one and one and one and one and one and one and one and one?" It appears that addition is intrinsically related to counting. Quantities are described either verbally or by number symbols (known as numerals), while variables (or generic names) substitute for unspecified quantities. The only thing that matters is the quantities involved. Why do we need special terms like addition and commutativity and symbols like "+" and "="? Because of the convenience of having short descriptives for very common situations and to underscore the generality of our understanding: the nature of objects in the groups is absolutely inconsequential. In mathematical terms, a + b = b + a - commutativity of addition. From the First Principle of Counting it follows that if we start counting with the second group and then move on to the first one, the result will be exactly the same. To obtain the total quantity in the two groups, we agree to first count one group and then continue counting the remaining one. But who said that mathematics is difficult? From the First Principle of Counting (sic!), this operation does not change the totality of objects in the two groups. Remove an object from one group and place it into the other one. What can be deduced from this statement? Assume we are given two groups of objects. The result of counting objects in a group does not depend on the manner in which the process of counting is conducted. Here then the First Principle of Counting This is because from this fact alone one may deduce a variety of conclusions. By merely verbalizing it we make a first step to understanding what mathematics is all about. We often count them a couple of times to see that we got the right number. However we go about counting the number of eggs in a basket the result is always the same. What makes counting possible? A simple fact that such a value exists. The purpose of counting is to assign a numeric value to a group of objects. The main property of counting is so fundamental to our perception of quantity that it is seldom enunciated explicitly. Counting various quantities is the foremost human activity in which children engage beginning at a very tender age.
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