Ответьте на вопросы

1) What were numbers initially used for?

2) What numbers were used for calculation?

3) What numbers are called whole numbers?

4) Why is the idea of negative numbers abstract?

5) What is a fraction?

6) What numbers does it contain?

7) What does the denominator mean?

8) What does the numerator tell us about?

9) What is the meaning of the word ²rational²?

10) What numbers are called rational?

11) Why do you think fractions are necessary?

12) What do rational numbers consist of?

13) What kinds of fractions do you know?

14) How can all rational numbers be represented?

15) Why can we think of natural numbers, whole numbers, and integers as subsets of the rational numbers?

16) What numbers cannot be expressed as a fraction?

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Irrational numbers The history of fractions The emergence and development of the concept of number and number theory Living numbers

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1. Прочитайте текст и озаглавьте его, выбрав один из предложенных ниже вариантов. Обоснуйте своё мнение пользуясь Приложением 1.

A. A placeholder

B. The inconceivable adventures of zero.

C. Zero

D. The idea of zero

The story of zero is the story of an idea that has aroused the imagination of great minds across the globe. Understanding and working with zero is the basis of our world today; without zero, we would lack calculus, financial accounting, the ability to make arithmetic computations quickly, and especially in today’s connected world, computers.

Zero was independently invented only three times. The first recorded zero is attributed to the Babylonians, the 3^rd century BC. A long period followed when no one else used a zero placeholder. Then the Mayans, halfway around the world in Central America, independently invented zero in the 4^th century CE. The final independent invention of zero in India was long debated by scholars, but seems to be set around the middle of the 5^th century. It spread to Cambodia around the end of the 7^th century. From India it moved into China and then to the Islamic countries. Zero finally reached Western Europe in the 12^th century.

About 3,000 years ago, when people started writing bigger numbers like "42" they had a problem: how to tell the difference between "4" and "40"? Without the zero, they look the same!

So they used a "placeholder", a space or special symbol, to show "there are no digits here". (So "5 2" meant ²502² (5 hundreds, nothing for the tens, and 2 units.)

The idea of zero had begun, but it wasn’t for another thousand years or so that people started thinking of it as an actual number. It was the Indians who began to understand zero both as a symbol and as an idea.

The Babylonians had a sexagesimal number system, that is, they counted in 60s, as we count in tens. They used two symbols to represent all the numbers from 1 to 59. The wedge was used for a one and the crescent equaled a 10. By grouping them together, they created symbols for all 59 numbers.

It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6. Babylonian mathematicians used the separator, a sort of sideways, superscript, double wedge (effectively the first zero) in the middle position only.

The Mayans had a very complex calendar system and needed a placeholder in their elaborate date system. It is the formal Long Count calendar that brought about the zero. The zero was often represented by a shell shape.

The final independent invention of the zero was in India. Math problems were written in verse and could be easily memorized, chanted or sung. Each word in the verse corresponded to a number. Brahmagupta, around 650 AD, was the first to formalize arithmetic operations using zero. He used dots underneath numbers to indicate a zero. These dots were alternately referred to as ²sunya², which means empty, or ²kha², which means place. Brahmagupta wrote standard rules for reaching zero through addition and subtraction as well as the results of operations with zero. The only error in his rules was division by zero, which would have to wait for Isaac Newton and G.W. Leibniz to tackle. It is recorded, that a dot or an open circle was used as zero before 628.

The Hindus influenced the numeration of nearby locales, and introduced the zero to the Chinese and to the Arabs who developed the modern day shape of numerals and passed them, along with zero, to the Europeans in the 12^th century.

Zero reached Baghdad by 773 AD. In the 9^th century, the Arabian mathematician Mohammed ibn-Musa al-Khwarizmi was the first to work on equations that equaled zero. He also developed quick methods for multiplying and dividing numbers known as algorithms (a corruption of his name). Al-Khwarizmi called zero ²sifr², from which our cipher is derived. By 879 AD, zero was written almost, as we now know it, an oval but in this case smaller than the other numbers.

And thanks to the conquest of Spain by the Moors, zero finally reached Europe; by the middle of the 12^th century, translations of al-Khwarizmi’s work had weaved their way to England.