Primary mathematics refers to mathematics that developed in the age of classical civilization, starting around the fifth century bc, and finally creating calculus at the end of the 17th century in newton and lebnitz, so it was more than 300 years ago. It's the math we learned from primary school to high school, which is the math that everyone has to learn. On the contrary, higher mathematics — the mathematics that has been developed for almost 300 years — may not be touched by some people in their lifetime. “if mathematics, as it was known before the seventeenth century, is referred to as primary mathematics, then it should be said that primary mathematics is insignificant compared to the mathematics that was created since then. In fact, one has the knowledge of newton's peak period and cannot be considered a mathematician today. Because, contrary to the general view, it is time to say that mathematics begins with calculus, not ends with it” (krein: mathematics in western culture, p19). In other words, primary mathematics is just normal life math. If you read science, elementary mathematics is definitely not enough. But anyway, elementary mathematics is the foundation。
According to the traditional division, primary mathematics is divided into three parts: algorithms — study of integers and fractions and their operation; algebras — use abstract symbols to represent mathematical objects, such as algebras, equations and functions, and study their operation and their rules; geometry — study of spatial graphics, points, lines, curves, and internal patterns of graphics。
Mathematics, of course, starts by counting, “as the exchange of goods evolves, people need to calculate the number of sheep ... How to write the numbers effectively is a problem that has been bothering humanity” (berlinhoff: this is a good mathematical history, p. 76). Counts require symbols, more so the idea of place. There is a practical basis for the notion of entry, such as that five fingers mean one hand. The difficulty is how to move forward in a uniform manner and express it succinctly. “our current method of writing figures is called the arabic count system. The system was invented by indians some time before 600 b. C. And, after centuries of refinement, was learned and applied by arabs when islamic forces expanded to india in the seventh and eighth centuries. Then europeans learned the system from arabs. (and eventually to the world). The system is characterized by the use of a bitmap and is based on 10 gills. Its basic symbol — 0,1,2,3,4,5,5,7,8,9 — is known as “arabic numerals”, indicating numbers ranging from zero to nine” (berlinhhoff: this is a good mathematical history, p81)。
Thus, we know that the math we learn is not “single” mathematics, but rather a more widespread and convenient “under contract”. So that we can accept another kind of computer-serving math — binary; we can also understand that the numbers that we still see in everyday life (such as roman numerals) and the evolutionary system (such as 60-digit clocks) are also mathematical。
Counting is then done. The concept of calculation is universal, and one plus one is always equal to two. But there is also the difficulty of how simple and clear the calculations can be. “the arithmetical symbols appear in writing at the beginning of the renaissance, but they are almost inconsistent between human beings and nations. With the invention of the 15th century print print, printed books began to show more consistency. However, it took a long time before the symbols we use today became a common component of written arithmetic” (berlinhoff: this is a good mathematical history, p85)。
More important than formal unification, calculations have contributed to the deepening of two mathematical concepts: the widening of the digital concept and the abstraction of the operational significance。
First, of course, the first count is just a “positive integer”, also called natural, very “natural”. The addition of natural numbers is not a problem, but the calculation is a problem if a large natural number is used to reduce natural numbers. One solution would be to reject such an operation, which would be considered unworkable. Another option was to broaden the notion of numbers, so zero, negative, fractional, unreasonable and even false. The acceptance of these new numbers is, first and foremost, a widening of the range of calculations, so that we can multiply or multiply by four the number of any number that is accessible, and extend both the multiplier and the opening。
But it's more important to show more of the math: naturally there is a deeper pattern structure than it seems, and mathematics is precisely what helps humans reveal and understand it. “1” means an apple, “three quarters” means that after an apple is divided into four parts, you get three of them, which is exactly the same as a “half” apple, or “one half”. But what does root 2 mean? Is it just an operation? No, that's exactly the length of the diagonal line in the square with a square length of 1. And the magic platinum is the length of the circle divided by the diameter. We thought mathematics was just a summary of practice, but now it is discovered that math sometimes goes ahead and waits for the ultimate appearance of a natural structure。
It's an abstract thing to say. The original calculation was very straightforward, plus consolidation, and your five sheep and my three sheep, of course, eight sheep. Multiplier is a simple addition, five times three is three five sheep, of course 15 sheep. As the notion of numbers widens, the meaning of computing becomes less and less understood. What does it mean to be small minus large? Negative minus negative? And the whole number multiplied by the fraction? And the fraction multiplied by the fraction? And the negative number multiplied by the negative number? Why does it become positive
The answers to these questions can be taken in two positions. One is to give up the question of meaning and fully understand the calculation as a rule, and you can follow the rules. This is the kind of "mathematics" that we are often told about -- mathematics is a logical rule system; and second, to keep meaning, you find again that mathematics, while sometimes taking a step ahead, actually has a natural structure. In other words, some sort of operation that looks like "unjustifiably" reveals a particular structure of nature. It's just that we sometimes don't understand it, or it's too complicated to understand it at once, because we don't have a high-level math base。
The medium-translation " algebra " can be understood literally as " alphanumeric replacements " , with mathematics moving from decimal algorithms (to numbers) to algebras (to letters), and mathematical algorithms (3+5 = 8) to algebras (a+b = c). The symbolicization of numbers is a major improvement in mathematical thinking, "the role of good mathematical symbols goes far beyond efficient stenography. Ideally, it should be a universal language capable of clarifying ideas, revealing patterns and providing generalizations. ... Today's algebra symbol is close to this ideal state, but its development is long and slow, and sometimes regressive” (berlinhoff: this is a good mathematical history, p132-133)。
The second step that the algebra brings to mathematical thinking is that we have to learn to understand something between "change and change." the letters in the algebra represent an uncertain fixed amount, expressed in mathematical language as " variable constants, not variables " . “variance” means a number that can represent any given range, such as any integer. But it is constant, and algebras can operate like an algorithm, in which letters are considered to be constants like a specific number, rather than variables in functions or unknowns in equations. For example, “-a” means a constant constant, but it is not necessarily negative because a may be negative in itself; b/a is a dichotomy algorithm that needs to be qualified immediately, otherwise it may become meaningless。
The core of the traditional algebra is the equation. From a historical perspective, algebra also evolved from an equation. “when we apply mathematics to the real world, the problem of solving equations naturally arises. It is no surprise, therefore, that almost everyone who learns mathematics, from egyptian copywriters to chinese civil servants, is looking for solutions to these problems” (berlinhoff: this is a good mathematical history, p140)。
The equation reflects the “practical” side of mathematics, ranging from application to relativity. The equation also fully demonstrates the “magic” of mathematics as a tool, and once the binary group is mastered, traditional dilemmas like chicken rabbit coops are no longer difficult. But the equation best reflects two characteristics of mathematics: logical expansion and deeper integration。
The simplest equation is a one-dollar one-time equation, and "dollar" refers to the number of unknowns and the number of unknowns. On this basis, logical expansion begins. To expand by a single number, there is a binary equation, a three-dollar equation, which leads to a single equation. Extending by “numbers” means a binary equation and three equations of one dollar, so that the equation is one dollar n. In combination, the family of multiple equations is constructed: binary equations, five-dollar equations three times, to n-n equations。
With an entire family, we can identify a number of key small branches, such as the “linear” character of all equations — a universal and important model of the real world. As a result, the mathematical study of linear equations has become a defining tool for solving such practical problems. The distinction between “linear” and “non-linear” issues remains useful today. We apply it not only to equations, but to many other issues” (berlinhoff: this is a good mathematical history, p144). Algebras have also evolved from primary algebras to upper algebras, i. E. Algebras with a platter and matrix as their core。
The fundamental problem of the equation is to solve it, and it's like playing magic. You can get a one-off result with little intelligence or experience, and you have to start over every time. But mathematics seeks a kind of “deep” unification, a universal root formula. The best expression of the character of mathematics is the history of solvency of the one-dollar-high equation. First, the mathematician weda gave the root formula of the binary equation, and then the 16-17 century european “arms race” for a higher equation (see berlinhoff: this is a good mathematical history, next chapter 11). Finally, in 1545, cardano “found a complete solution to all three equations. His assistant rodovico-farraly applied the same point of view to the equations in general four times and managed to find a way of understanding. The next goal is five equations. This proved much more difficult. In fact, it is impossible to find a formula that solves the five equations in general. This fact prompted the generation mathematicians to start asking deeper questions. Gradually, the theory of multiplicity and its roots has evolved” (berlin hof: this is a good mathematical history, p155, 157, 48)。
The third area of primary mathematics is geometry. As stated earlier, from a historical point of view, geometry predates algebra, a specialty and treasure of the ancient greeks. The treasure is naturally the geometry original of euclid. “the book is divided into 13 volumes, containing a total of 465 “problems” (which we can now call the inference), each of which is evidenced by the previous statements. A simple start (all based on one) — 23 definitions, 5 common concepts and 5 assumptions — euclid has re-established the entire geometry of the plane. His writings were so comprehensive and clear that since his time, geometry had become a widely accepted source of information for the study of geometry on the plane. Even today's geometry in high school (lower secondary school) has been largely adapted from the geometrics of euclid” (berlin hof: this is a good mathematical history, p179,181)。
“the geometry original is not just a discussion of shapes and numbers, but a lesson in how to think! Teach you how to use logic to think about anything — how to build a complex theory step by step, and everything new is firmly connected to what has been built. For more than 2,000 years, the geometry of euclid flats has shaped western thinking. Indeed, many of the most influential works in politics, literature and philosophy cannot be truly understood without your appreciation of euclid” (berlin hof: this is a good mathematical history, p181)。
“high school” elementary mathematics (theoretical geometry, stereogeology, function theory, etc.) has been largely detached from regular mathematics (mathematics of daily application), which is actually a pre-studies of advanced mathematics (mathematics of technological application). So in some western countries, high school mathematics is no longer a compulsory subject, but rather an optional subject of different grades chosen by future universities。
