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  1. Перепишіть речення і перекладіть рідною мовою, зважаючи на різні значення підкреслених слів as, one it, that, but,for.

5.1. That the trimmers used in planning mill are of about the same construction as those used in the sawmills is a well-known fact.

5.2. If the valve in the cylinder moves too easily it should be replaced immediately.

5.3. The simplest kind of lever is one in which the arms are of equal length.

5.4. He was surprises as if he had never seen such a device.

5.5. Nothing but a double row of ball bearings can improve the machine.

  1. Перекладіть англійською мовою.

6.1. Цей цикл продовжується в кожному з циліндрів.

6.2. Лазери, що були щойно описані, не єдині з наявних.

6.3. Нас повідомили, що обладнання вже було доставлено.

6.4. Вчений сказав, що цей цикл буде продовжуватись.

6.5. Вони не знали, що на них вже чекали.

  1. Перепишіть текст та перекладіть його рідною мовою.

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere. Mathematicians formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.

There is debate over whether mathematical objects exist objectively by nature of their logical purity, or whether they are manmade and detached from reality. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions". Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."

Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.

The major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e., arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.