13 października 2015

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Vocabulary

Exercises

Exercise 1. Match the words in column A with the words in column B.

Exercise 2. Answer the following questions.

1. What are natural numbers?
2. What are natural numbers sometimes called?
3. What are words used for counting called in common language?
4. What are words used for ordering called in common language?
5. What other number sets may be built on the basis of the natural numbers?
6. Is 0 included in the set of natural numbers?
7. What symbol is usually used to refer to the natural numbers?
8. What algebraic properties do the natural numbers have? Can you define them?

Exercise 3. Listed below are algebraic properties of real numbers. Write sentences describing them as shown in the example.

Example: Closure under addition: a + b; natural number. → For all natural numbers a and b, a + b is a natural number.

1. Closure under multiplication: a × b; natural number
2. Associativity under addition: a + (b + c) = (a + b) + c
3. Associativity under multiplication: a × (b × c) = (a × b) × c
4. Commutativity under addition: a + b = b + a
5. Commutativity under multiplication: a × b = b × a
6. Identity: a + 0 = a; a × 1 = a
7. Distributivity of multiplication over addition: a × (b + c) = (a × b) + (a × c)
8. Addition property of equality: if a = b, then a + c = b + c
9. Multiplication property of equality: if a = b, then ac = bc

Exercise 4. Solve the following equations.

• x + 4 = 16
• x – 8 = 21
• y × 6 = 36
• 12 + 4 =
• 49 ÷ 7 =
• 17 × z =234
• 2 + 2 =
• 11 – p = 12
• 13 × q = 130
• 12 ÷ z = 4

Exercise 5. Translate the second part of the following sentences into English.

There is exactly…

• jedna liczba naturalna n, taka że n − 2 = 4.
• jedna liczba całkowita x, taka że x × 3 = −6.
• jedna liczba wymierna y, taka że y + 8 = 8⅕.
• jedna liczba naturalna z, taka że z ÷ 4 = 4.
• jedna liczba rzeczywista p, taka że p × 5 = 25.

Exercise 6. Translate the following sentences into English. Use the expression „studied in”.

1. Podzielność i rozkład liczb pierwszych są przedmiotem badań teorii liczb.
2. Rozkład i uporządkowanie liczb naturalnych są przedmiotem badań kombinatoryki.
3. Liczby naturalne są przedmiotem badań teorii liczb.
4. Zbiory liczb są przedmiotem badań teorii zbiorów.
5. Liczby są przedmiotem badań matematyki.
6. Liczby rzeczywiste są przedmiotem badań teorii liczb.
7. Liczby zespolone są przedmiotem badań teorii liczb.
8. Właściwości liczb wymiernych są przedmiotem badań teorii liczb.

Exercise 7. Change the following sentences into the passive voice as in the example.

Example: You use natural numbers for counting. → Natural numbers are used for counting.

1. You use cardinal numbers for counting.
2. You use ordinal numbers for ordering.
3. You use N to refer to the set of all natural numbers.
4. You use real numbers for measurement.
5. You use digits to represent numbers.
6. You use natural numbers to express the size of a finite set.
7. You use ordinal numbers to describe the „size” of a well-ordered set.
8. You use real numbers to measure continuous quantities.
9. You use rational numbers to express fractions.
10. You use the + operator for addition.
11. You use the − operator for subtraction.
12. You use the × operator for multiplication.
13. You use the ÷ operator for division.
14. In computer programming you use the slash (/) for division.

Exercise 8. Jesteś leśniczym i polecono ci zasadzić 10 drzew w pięciu rzędach po cztery drzewa. Jak to zrobisz? Rozwiązanie możesz wpisać w komentarzu na dole strony.

Exercise 9. Fill in the blanks with the words or expressions provided below.

natural whole counting ordering cardinal ordinal from sets extension on properties distribution numbers theory combinatorics with to non positive countably

A.
………. numbers (sometimes called the ………. numbers) are those used for ………. and ……….. In common language, words used for counting are „………. numbers” and words used for ordering are „………. numbers”.

B.
The natural numbers are the basis ………. which many other number ………. may be built by ……….: the integers, the rational numbers, the real numbers, the complex numbers, and so ………..

C.
………. of the natural numbers, such as divisibility and the ………. of prime ………., are studied in number ……….. Problems concerning counting and ordering are studied in ………..

D.
Some mathematicians begin the natural numbers ………. 0, corresponding to the ……….-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the ………. integers 1, 2, 3, …. The set of all natural is ………. infinite: it is infinite but countable by definition.

Exercise 10. Translate the text into Polish.

Exercise 11. Summarize the text orally using questions from exercise 2 as a plan. Practice until you can say the summary without looking at the text.

Exercise 12. Watch the video from the beginning of the lesson again and answer the following questions.

1. What three types of numbers does the author describe?
2. What other thing is important in algebra?
3. What does infinity mean?
4. What’s the difference between natural and whole numbers?
5. What are the short bars crossing the number line called?
6. What is the general subject matter of the series of videos?
7. What does the expression "stay tuned for lesson two" mean?