Download Lekcja 2 A

SJO PW - Język angielski ogólnotechniczny, Poziom B2
Opracowanie: Teresa Olechowska
Lekcja 2 A
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MATHEMATICS
INTRO:
NUMBERS AND OPERATIONS
TASK 1
Think and try to answer the following questions:
1. What is counting?
2. What in fact are numbers and basic operations?
TASK 2
Can you read the following number 8, 375, 486, 493?
8, 375, 486, 493
|
|
| thousand
|
| million
| billion
Let’s start from the smallest one:
3 = three; 93 = ninety three; 493 = four hundred (and) ninety three;
let’s add thousands (a comma):
6, 493 = six thousand four hundred (and) ninety three;
86, 493 = eighty six thousand four hundred (and) ninety three;
486, 493 = four hundred (and) eighty six thousand four hundred (and) ninety three;
let’s add millions (the second comma):
5, 486, 493 = five million four hundred (and) eighty six thousand four hundred (and) ninety three;
75, 486, 493 = seventy five million four hundred (and) eighty six thousand four hundred (and) ninety three;
375, 486, 493 = three hundred (and) seventy five million four hundred (and) eighty six thousand four hundred (and)
ninety three;
and finally let’s have some billions (the third comma) = milliards – nowadays even in Great Britain:
8, 375, 486, 493 = eight billion three hundred (and) seventy five million four hundred (and) eighty six thousand four
hundred (and) ninety three;
NOTE that WE DO NOT USE PLURAL of thousand, million, and billion in case of expressions such as:
‘two thousand miles’, ‘two million ways’, etc.!
Now try to read the whole number as quickly as you can.
If you wish to practice this, have a look at home at this very short lesson:
http://www.youtube.com/watch?v=1LDmrYpaQkc
INPUT - HOMEWORK
TASK 3
Answer the following questions:
1. What do we call an operation, when we ADD one quantity TO another, using the symbol + (plus) and the
result is called the SUM?
………………………
2. What do we call an operation, when we SUBTRACT one quantity FROM another, using the symbol – (minus)
and the result is called the DIFFERENCE?
………………………
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SJO PW - Język angielski ogólnotechniczny, Poziom B2
Opracowanie: Teresa Olechowska
Lekcja 2 A
______________________________________________________________________________________________
3. What do we call an operation, when we MULTIPLY one quantity BY another, using the symbol x (multiplied
by or times) and the result is called the PRODUCT?
………………………
4. What do we call an operation, when we DIVIDE one quantity BY another, using the symbol ÷ (divided by)
and the result is called the QUOTIENT?
………………………
NOTE that results of these operations are indicated by the symbol = (equals).
TASK 4
FRACTIONS VULGARS AND DECIMAL
Do you know how to read fractions – vulgar and decimal ones?
Try with the following example:
4/5
And with this one:
2.25
In the first case it is: four fifths or four over five;
In the second – is: two point two five
NOTE that:
a. in Polish we mark decimals with a comma
b. e.g., 17 ÷ 3 = 5.67 correct to two decimal places
TASK 5 (optional)
Thus:
are numbers ‘discovered’ or ‘invented’?
Consider the following examples:
1. 
the ratio of the circumference of a circle to its diameter
It equals 3.141582… (an infinite, non-recurring decimal).
2. e
the base of natural logarithms e = 2.7182818…
It can be defined as the sum of the infinite series.
3. -1
When negative numbers were introduced from the east many considered them ‘occult’ and
Girolamo Cardano referred to them as ‘fictitious’. After all, who has seen ‘minus one cow’?
4. i
the square root of –1, which is neither positive nor negative
Incidentally mathematicians, like physicists, enjoy giving things totally inappropriate names. Thus  and e are
referred to as ‘irrational’ and ‘transcendental’ numbers while i is ‘imaginary’.
TASK 6
Match the following sets of numbers with their explanation:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NATURAL NUMBERS
INTEGERS
NEGATIVE INTEGERS
NON-INTEGERS
RATIONAL NUMBERS
IRRATIONAL NUMBERS
REAL NUMBERS
DECIMALS
IMAGINARY NUMBERS
COMPLEX NUMBERS
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SJO PW - Język angielski ogólnotechniczny, Poziom B2
Lekcja 2 A
Opracowanie: Teresa Olechowska
______________________________________________________________________________________________
a. cannot be written as a fraction a/b, where a and b are integers and b is not zero (√2 = 1.414…; π = 3.14159265…);
they cannot be represented as terminating or repeating decimals
b. natural numbers preceded by the minus sign
c. a + bi; where a and b are real numbers, and i is imaginary one
d. fractions, where both the denominator and the numerator are integers
e. whole numbers = (…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…)
f. counting numbers = (1, 2, 3, 4,…)
g. usually expressed by a real number and a imaginary number; cannot be represented as points on a number line
h. rational numbers + irrational numbers
i. integers + non-integers
j. fractions with a denominator of 10, 100, 1000, etc. (3/10=0.3; 17/100= 0.17. etc.)
TASK 7
Match the numbers with their descriptions, e.g.:
4/7 – positive rational non-integer
1.
2.
3.
4.
5.
6.
7.
8.
negative rational non-integer
complex number
imaginary number
positive non-integer
negative integer
negative irrational
positive or negative irrational - real number
positive integer – natural number
a.
b.
c.
d.
e.
f.
g.
h.
√5
27
–11/12
132 + 7i
0.33
√-23
-√4
-
TASK 8
Are the following statements true or false:
1.
2.
3.
4.
5.
If m and n are integers, m/n is a rational number.
The set of irrational numbers includes negative integers.
Neither irrational numbers nor complex numbers may be represented as points on a number line.
The symbol i represents a complex number.
A complex number consists of at least two parts.
TASK 9
FORMULAE AND EQUATIONS
An EQUATION is a mathematical statement that asserts the equality of two expressions, e.g., x + 9 = 21.
An expression which contains a square as the highest power of any letter (x², y², etc) is called quadratic, and if such an
expression is equal to some value, the resulting equation is called a quadratic equation.
As in the Pythagorean Theorem:
a² + b² = c²
When a particular problem is solved then the method of solving is reduced to a fixed pattern, written down as a
FORMULA. Thus a formula is a rule or principle frequently expressed in algebraic symbols.
For example, the statement “Average speed is equal to the distance covered divided by the total time taken”, can be
written as the formula:
D
S = --T
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SJO PW - Język angielski ogólnotechniczny, Poziom B2
Lekcja 2 A
Opracowanie: Teresa Olechowska
______________________________________________________________________________________________
or the famous Einstein’s equation as:
E = mc²
Have a look at the following symbols:
0<x<1
0≤x≤1
x²
x³
x
xⁿ
√x
³√x
√x
ⁿ√x
(x + y)²
(x/y)²
n!
∞
zero is less than x is less than 1
zero is less than or equal to x is less than or equal to 1
x squared
x cubed
x to the fourth/x to the power fourth
x to the n/x to the nth/x to the power n
root x/ square root x/ the square root of x
cube root x
fourth root x
nth root x
x plus y all squared
x over y all squared
n factorial/ factorial n
infinity
is proportional to…/ varies as…
∑
sum over… from… to… of
∫
the integral from… to…of with respect to…
means a1 + a2 + … + an;
FOR MORE visit the following websites:
http://en.wikipedia.org/wiki/List_of_mathematical_symbols
or
https://www.google.pl/search?q=mathematical+symbols&hl=pl&rlz=1G1GGLQ_PLPL314&prmd=imvns&
tbm=isch&tbo=u&source=univ&sa=X&ei=vaK6T82gJevP4QT4gIHACQ&ved=0CGkQsAQ&biw=1530&b
ih=692
Knowing the above, how would you rewrite in symbols the following equation:
1. x plus nine equals twenty three
………………………………
2. twelve plus four a equals seven a minus twenty-one
………………………………
3. x squared minus five x plus six equals zero/nought (BE)
………………………………
4. five c plus six equals two c plus twenty-four
………………………………
5. twelve minus two b equals four b plus thirty-six
………………………………
[Optional] Now, solve the above equations.
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SJO PW - Język angielski ogólnotechniczny, Poziom B2
Opracowanie: Teresa Olechowska
Lekcja 2 A
______________________________________________________________________________________________
CLASSWORK
TASK 10
GAMELAND
This is like a crossword puzzle, except you fill it with numbers instead of letters.
1
2
3
4
5
6
7
8
9
10
ACROSS:
1. a score
2. the next number in: 1, 3, 6, 10, 15, …
3. XLIX in Arabic numbers
4. the next number in: 1, 4, 9, 16, 25, 36, 49, …
6. three times itself three times over
8. any two figures from 5 DOWN
9. two dozen
10. binary for ‘2’
DOWN:
1. two times itself eight times over
2. seventeen times seventeen plus three
5. 37 dozens (all figures are the same)
7. numbers of degrees in two full turns (look at a protractor!)
1.
2.
3.
4.
5.
6.
7.
BIBLIOGRAPHY
A Mathematical Dictionary, Jason, R.E., Pergamon Press, 1979
Alex’s Adventures in Numberland, Bellos, Alex, Bloomsbury, 2010
Computer Adventures in Lingua Land, Olechowska, Teresa, Rouba, Wojciech, WSiP, 1992
English for Basic Maths, Blackie, David, Nelson, 1978
Lingua Land Kappa Land, Olechowska, Teresa, WPW, 1986
Mathematics and the Imagination, Kasner, Edward, Newman, James, Penguin Books, reprinted 1979
MINI anglojęzyczne – EAP & ESP, Olechowska, Teresa, wydanie własne, 2012
http://dictionary.reference.com
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