Rational and irrational numbers

1)… is a theory of measurement. The 2) … for measuring numbers are the rationals and irrationals. But let us start at the beginning.

The following numbers of arithmetic are the counting-numbers or, as they are called, the 3) … numbers: 1,  2,  3,  4,  and so on. If we include 0, we have the 4)… numbers: 0,  1,  2,  3,  and so on. And if we include their algebraic negatives, we have the 5)… : 0,  ±1,  ±2,  ±3,  and so on. ± ("plus or minus") is called the double sign. The following are the square numbers, or the 6) … : 1   4   9   16   25   36   49   64, and so on. They are the numbers 1· 1,  2· 2,  3· 3,  4· 4,  and so on.

A rational number is a number that can be written as a simple 7) … (or as a ratio). 5 can be represented as a 8) … or a quotient of 5 and 1 and so on. But the square 9) … of 2 cannot be written as a simple fraction. And there are many more such numbers, and because they are not rational they are called irrational.

The ancient Greek 10)… Pythagoras believed that all numbers were rational in other words could be written as a fraction, but one of his students Hippasus proved (using geometry, it is thought) that you could not represent the square root of 2 as a fraction, and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect squares. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!

But nowadays we know that a rational number is a number that can be expressed as a fraction p/q where p and q are 11) … and  q is not equal to 0. A rational number p/q is said to have 12) … p and 13) … q. Numbers that are not rational are called 14)… numbers. The real line consists of the 15) … of the rational and irrational numbers.

The set of all rational numbers is referred to as the "rationals," and forms a field that is denoted   . Here, the symbol   derives from the word 16) …, which means "the result of division", "ratio," and first appeared in Nicolas Bourbaki's "Algebra".

Examples of rational numbers include  , 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of 17) … all rational numbers systematically.

Task 8

Translate into English:

1. Раціональні та ірраціональні числа необхідні нам для вимірювання предметів та речовин. 2. Квадрат цілого числа – це добуток від числа, помноженого на те саме число. 3. Показник степеня вказує на те, скільки разів слід помножити певне число на таке саме число. 4. Раціональне число – це число, що може бути представлене у вигляді простого дробу. 5. Вчитель попросив учнів довести теорему Піфагора. 6. Скільки способів доведення теореми Піфагора вам відомо? 7. Піфагор не зміг спростувати існування ірраціональних чисел. 8. Чим відомий Піфагор? Це видатний математик стародавньої Греції. 9. Простий дріб складається з чисельника та знаменника. 10. Знаменник вказує на те, на скільки частин поділене ціле число. Чисельник вказує на те, скільки таких частин ми взяли.

Task 9

Tell the class about rationals and irrationals using the following list as promts:

Measurement Necessary Counting numbers Whole numbers Integers Double sign Square numbers Exponent Simple fraction Square root of 2 Irrationals

Pythagoras Hyppasus proved Accept existence Thrown overboard Numerator Denominator Real line Set Field Q Farey sequence

Task 10

Tell the class about rational and irrational numbers using the question list to the text as a plan.

Task 11

Prepare the reports on the following topics and be ready to present them in class:

1. Nicolas Bourbaki – a real or imaginary person?

2. What we know about Farey sequence and its author.

3. Pythagoras and his discoveries.

Task 12

Read the text and render it in English.