Download I Numbers and Mathematical Expressions in English

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UNIVERSITÄT HANNOVER
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ENGLISCH
I
Dermot McElholm
Numbers and Mathematical Expressions in
English
Am Judenkirchhof 10 - 30167 Hannover - Tel.: 0511/762-4914
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USING NUMBERS
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1.1 Numbers in English
1.1.1 Writing the numbers
1.1.2 The ordinal numbers
1.1.3 Fractions
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1.2 DECIMALS
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1.3 NUMBER SYSTEMS
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1.4 PERCENTAGES AND UNITS
1.4.1 Percentages
1.4.2 Units
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MATHEMATICAL EXPRESSIONS
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2.1 Trigonometric functions
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2.2 Derivatives and integrals
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2.3 Exponential functions and logarithms
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Numbers and mathematics
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NUMBERS & MATHEMATICAL
EXPRESSIONS
1 USING NUMBERS
1.1 Numbers in English
Let us look at various aspects of expressing numbers in English, starting with
how to write the cardinal numbers.
1.1.1 Writing the numbers
There are several easy rules for writing the cardinal numbers, as the
following series show:
• twenty-one (21), thirty-three1 (33), ninety-five (95)
• a/one hundred and thirty-nine (139), two thousand, four hundred and
seventy-two (2,472)
• two hundred and three (203), five thousand and ten (5,010)
The tens and the ones are connected with a hyphen. Otherwise numbers are
written separately. We write and between hundreds and tens, hundreds and
ones, thousands and ones, in other words, between any higher units than tens
and lower units which can either be tens or ones, but not between thousands
and hundreds or millions and thousands; note that the and always comes
before the last unit.
The numbers are usually invariable: we say two hundred for example and
not *two hundreds. The same applies to million and billion. The latter has the
general meaning nowadays in English-speaking countries of 109 (in Britain it
used to mean 1012). The word trillion is also used, meaning 1012.
We only use the plural when we do not mean a definite number, e.g. the
storm caused several billions of dollars of damage, millions of years ago.
Every three digits to the left is separated by a comma or (a more recent
development and used in some journals) by a space (but not by a decimal
point as in German):
1 Remember the difference in stress between 'thirty and 'thirteen.
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• 3,456 (3 456)
• 123,456 (123 456)
• 123,456,789 (123 456 789)
Note that years are generally expressed in English not with the hundreds or
thousands but as two separate two-digit numbers:
♦ nineteen ninety-eight (1998)
♦ ten sixty-six (1066)
♦ nineteen oh one (1901)
Exceptions are:
⇒ two thousand (2000)
⇒ two thousand and ten (2010)
Time will tell if we will also abbreviate these years (to twenty forty etc.) in the
coming millennium.
We usually put cardinal numbers before other adjectives:
∗ the four green fields
1.1.2 The ordinal numbers
The ordinal numbers are usually formed by adding -th, except for:
➣ first
➣ second
➣ third
and their combinations: forty-first. Note the irregular spelling in the
following:
☛
☛
☛
☛
☛
fifth
eighth
ninth
twelfth
twentieth, thirtieth etc.
Dates are written and expressed in the following ways:
♦ 19th July 1998 = the nineteenth of July nineteen ninety-eight (GB)
♦ July 19, 1998 = July nineteenth, nineteen ninety-eight (USA)
Larger ordinal numbers are written as follows:
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✪ 95th = ninety-fifth
✪ 101st = one hundred and first
We usually put cardinal numbers before other adjectives, even before ordinal
numbers:
■ the first three American presidents
1.1.3 Fractions
Fractions are rational numbers that can be expressed as the ratios of two
integers a/b: a half/one half, one third, two thirds, three quarters. The ordinal
numbers are generally used, with the following exceptions: half, quarter. A
hyphen is generally used, apart from fractions with a, e.g. a half, a third;
fractions are can take the plural.
Fractions in algebra may be expressed as in the form:
1+ x
1− x
This is expressed in words as: one plus x over [or: divided by]one minus x
1.2 DECIMALS
The following is an example of a decimal fraction:
♦ 0.0345
Note the following:
a)
b)
−
−
−
c)
we use a decimal point in English instead of a comma in German;
0 is generally represented by:
zero (more mathematical)
nought (general technical)
oh (everyday English)
we express each digit separately, so the number reads:
nought point nought three four five
OR
point nought three four five
The number above is given to four decimal places; it could be the number
0.0345267 given to (or rounded down to) four significant figures.
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1.3 NUMBER SYSTEMS
The set of real numbers has as its elements the rational numbers and the
irrational numbers. The former are the whole numbers or integers
–
0, +1, +2, +3,...
and non-integers, numbers that can be expressed as the ratio of two whole
numbers, i.e. the fractions above. The positive integers
⇒ 0, +1, +2, +3,...
are called the natural numbers.
Contained in the set of integers is the subset of prime numbers which are
defined as being only divisible by 1 and the number itself. The set of positive
prime numbers is:
∗ {1, 2, 3, 5, 7, 11, 13, 17, 19...}
An irrational number is a real number that cannot be written as an integer or
as a quotient of two integers. The irrational numbers are infinite, nonrepeating decimals, e.g. π.2
1.4 PERCENTAGES AND UNITS
1.4.1 Percentages
Percentages are generally expressed together with the preposition of and with
or without the definite article:
(5) It will be noted from the normal distribution table that approximately 68
percent of the area under the curve lies within the range z = -1 and z = +1 or
within +1 standard deviation from the mean...
(6) To determine P95, arrange the values in order of size and count off 95 per cent
of them. The value appearing nearest that point is the 95th percentile.
We use the definite article when we are saying something about a determined
or definite part of a whole, while we leave out the article when we are listing
the parts of a whole.
1.4.2 Units
2 This number is in fact a transcendental number.
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Most units in English are treated as normal countable nouns that take a
singular and a plural:
•
•
•
•
•
10 m
85,000 gal
35oC
40 kg
85 lb
= ten metres
= eighty-five thousand gallons
= thirty-five degrees Celsius
= forty kilograms (kilos)
= eight-five pounds
There are exceptions: one hundred bar.
However, when a unit is used to modify another noun, then it is written as
follows: a ten-degree rise in temperature. Here the expression ten-degree
modifies the noun rise; it is written in the singular and joined together with a
hyphen.
Note the following: m² = square metres, m³ = cubic metres, and units such
as:
➲ N/m2 = newtons per square metre
The Americans still use the old British units: gallons, feet, pounds –
abbreviated to gal, ft, lb. The following conversions apply:
a)
b)
c)
d)
1 US gallon = 3.7854 litres (1 UK gallon = 4.546 litres);
1 inch = 2.54 centimetres
1 foot = 0.3048 m
1 pound = 453.592 g
Note the American spelling of the following units: liter, meter, kilometer
(British -re → American -er).
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2 MATHEMATICAL EXPRESSIONS
There is a wide variety of mathematical expressions in English. Here are some
of the most important.
2.1 Trigonometric functions
Let us start with the trigonometric functions. The Pythagorean theorem can be
stated as:
A2 + B 2 = C 2
where A and B are the sides of a right-angled triangle and C is the
hypotenuse.
The trigonometric functions are sine (sin), cosine (cos), tangent (tan), and are
defined as follows:
TABLE 1
sin α =
A
C
cosα =
B
C
tan α =
A
B
The law of Sines states:
sin α sin β sin γ
=
=
A
B
C
This reads: sine alpha over A equals sine beta over B...
2.2 Derivatives and integrals
A function of a single variable x is denoted by f(x), which reads f of x. The
derivative is:
dy
dx
and is expressed as dy by dx.
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An integral of f (x)dx is any function whose differential is f ( x ) dx , and is
denoted by
∫
f ( x )dx , which reads the integral of f of x dx. All the integrals
of f ( x ) dx are included in the expression
∫ f ( x)dx + C , where ∫
f ( x )dx is
any particular integral, and C is an arbitrary constant. The following integral
x1
∫ f ( x)dx
x0
is expressed as the integral from x zero to x one of f of x dx.
The following are some commonly used symbols in mathematics:
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Table 1: COMMONLY USED MATHEMATICAL SYMBOLS
Symbol
±
=
≡
≅
≠
>
<
≥
≤
−
÷
/
×
⋅
x2
x3
xn
Meaning
Plus or minus
Equals
Is identical with
Approximately equals
Is not equal to
Greater than
Less than
Greater than or equal to
Less than or equal to
Minus
Sign of division
Divided by
Times, by
Multiplied by
x squared
x cubed
x to the [power of] n
x [raised] to the nth
power
(the) square root (of)
√
Approaches, tends to
→
Varies as
∝
| |
Absolute value
Infinity
∞
n!
Factorial, n(n-1)(n-2)...1
Sum of a series of
∑
numbers
Therefore
∴
dx
Differential of x
Increment of x
∆x
Partial derivative
∂u/∂x
Nabla
∇
Thus we express the following equation
a 2 + b2 = c2
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in language as: a squared plus b squared equals c squared.
Note the following:
( a + b) 2
is expressed as: a plus b all squared.
In vector analysis we draw a distinction between two kinds of product,
the scalar or dot product and the vector or cross product. The scalar
product is defined as:
A ⋅ B = A B cos θ
while the vector product is
A×B
2.3 Exponential functions and logarithms
The basic result is that if y = ln x, that is y is equal to the natural logarithm of
x, y equals log n of x or y equals ln of x, then x = ey. The definition of the
natural logarithmic function y = loge x is expressed as y is equal to the
logarithm/log of x to the base e.
EXERCISE
A. EXPRESS THE FOLLOWING IN WRITTEN FORM:
1. 2,187
2. 4.03187
3. 412,397 km²
4. 0.001%
5. 1.602192 × 10-19
6. 20/09/1904
7. 7/8
8. 1.380622 x 10-23 J K-1
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B. EXPRESS THE FOLLOWING EQUATIONS IN WORDS:
1. ( x − a )² + ( y − b )² = r ²
2.
d ( e u ) = e u du
3.
x + iy = r (cosθ + i sin θ ) = re iθ
4.
a b = e b ln a
5.
d
dv
du
f (t )dt = f (v ) − f (u)
∫
dx u ( x )
dx
dx
v( x)