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UNIT 1 VOCABULARY: REAL NUMBERS
1.1. Rational Numbers
A rational number is any number which can be written in te form of a fraction. Rational
numbers are represented by the letter ℚ.
a
, where a and b are integers and b ≠ 0.
b
So, a rational number can be written as
But, what numbers can be expressed as a fraction?
Rational numbers include:
RATIONAL NUMBERS
Natural Numbers ℕ
6=
6
1
Integers ℤ
−3=−
ℚ
Exact (or terminating)
decimal numbers
3
1
0.75=
3
4
Recurring decimal
numbers, with a
periodic part.
1.6666...=
5
3
Exercise. Match each number with an appropriate sentence.
You have learnt to represent natural numbers, whole numbers an integers on a number line.
• Natural Numbers: the line extends indefinitely only
to the right side of 1.
•
Whole numbers: the line extends indefinitely to
the right, but from 0.
•
Integers: the line extends
indefinitely on both sides. Do you see
any numbers between –1, 0; 0, 1 etc.?
•
Rational Numbers: the line extends
indefinitely on both sides. But you can now see
numbers between –1, 0; 0, 1 etc.
Any rational number can be represented on the number line. In a rational number, the denominator
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tells us the number of equal parts into which the first unit has been divided; the numerator tells us
‘how many’ of these parts are considered.
4
7
So, a rational number such as
means four of nine equal parts on the right of 0, and for −
4
9
1
we make 7 markings of distance
each on the left of zero and starting from 0.
4
DENSITY
The set or rational numbers is dense, this is, between any two rational numbers, there is
always another rational number.
a +b
In general, if a and b are any different rational numbers, then
is a rational
2
number between them. This number is the average (or mean) of a and b, so it makes sense that it is
between them.
5
1
For example, if you want to find a rational number between
and
, you can do
12
2
5 1
5 6
11
+
+
12 2 12 12 12 11
=
=
=
2
2
2 24
Warning!!!
a +b
is not the only rational number between a and b. In fact, between any two distinct rational
2
numbers there are infinitely many other rational numbers.
1.2. Irrational Numbers
Irrational numbers are numbers that cannot be written as a fraction with the numerator and
denominator as integers.
FAMOUS IRRATIONAL NUMBERS
Pi is a famous irrational number. People have calculated Pi to over one million
decimal places and still there is no pattern. The first few digits look like this:
3.1415926535897932384626433832795 (and more ...)
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The number e (Euler's Number) is another famous irrational number. People have
also calculated e to lots of decimal places without any pattern showing. The first
few digits look like this:
2.7182818284590452353602874713527 (and more ...)
The Golden Ratio is an irrational number. The first few digits look like this:
1.61803398874989484820... (and more ...)
Many square roots, cube roots, etc are also irrational numbers. Examples:
√ 2 = 1.4142135623715 ... (etc)
√ 3 = 1.732050807568 ... (etc)
BE CAREFUL!
√ 4 = 2 (rational), and
√9
= 3 (rational) ...
so not all roots are irrational.
2.1. Real Numbers
Real Numbers include:
•
•
Rational numbers, and
Irrational numbers.
Exercise. Classify according to number type. Notice that some numbers may be of more than one type.
14
−
a) 1.01001000100001…
d)
7
b) 2.05
e) 3.157157157……
̂
c)
f)
−0.0 3
√−1
2.2. The Number Line
The Number Line is a line where you can represent every real number.
In fact a Real Number can be thought of as any point anywhere on the number line:
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2.3. Absolute Value
Absolute Value means how far a number is from zero:
"6" is 6 away from zero, and "-6" is also 6 away from zero, so the absolute value of 6 is 6, and the
absolute value of -6 is also 6.
So, in practice, "absolute value" means to remove any negative sign in front of a number, and to think
of all numbers as positive (or zero).
To show that you want the absolute value of something, you put "|" marks either side ("bars"):
For example:
|-5| = 5
|7| = 7
It doesn't matter which way around you do a subtraction, the absolute value will always be the same:
|8-3| = 5
|3-8| = 5
(8-3 = 5)
(3-8 = -5, and |-5| = 5)
2.4. Distance
The distance between two numbers a and b on a number line is the absolute value of their difference:
d(a,b) = |b – a| = |a – b|
For example, the distance between –3 and 4 is
or
d(4,-3) = |4 – (–3)| = |4 + 3| = |7| = 7
d(-3,4) = |-3 – 4| = |-7| = 7
2.5. Intervals
Interval: all the numbers between two given numbers.
For example, all the numbers between 2 and 4 is an interval.
The interval 2 to 4 includes numbers such as:
2.1
2.1111
2.5
2.75
7
2.80001
π
3.777777 ...
2
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You can represent intervals in three different ways:
•
Inequalities
With inequalities you use the following symbols:
>
≥
<
≤
Greater than
Greater than or equal to
Less than
Less than or equal to
Examples:
1<x<5
-2 ≤ x ≤ 1
x ≤ 20
x>0
•
➔ means all numbers between 1 and 5 (but including neither 1 nor 5)
➔ means all numbers between -2 and 1 (including -2 and 1)
➔ means all numbers less than or equal to 20.
➔ means all positive numbers.
Interval Notation
In "Interval Notation" you just write the beginning and ending numbers of the interval, and
use:
Symbol
Name
When do I use it?
( )
Round
brackets
If you don't want to
include the end value
[ ]
Square
brackets
If you want to include
the end value
Examples:
(5, 12]
(-5, -1)
•
➔ means all numbers between 5 and 12 (not including 5 but including 12)
➔ means all numbers between -5 and -1 (including neither -5 nor -1)
Number Line
With the Number Line you draw a thick line to show the values you are including, and:
▪
a filled-in circle if you want to include the end value, or
▪
an open circle if you don't
Like this:
means all the numbers between 0 and 20, do not include
0, but do include 20.
OPEN OR CLOSED?
The terms "Open" and "Closed" are sometimes used when the end value is included or not:
(a, b)
a < x < b
an open interval
[a, b)
a ≤ x < b
closed on left, open on right
(a, b]
a < x ≤ b
open on left, closed on right
[a, b]
a ≤ x ≤ b
a closed interval
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TO INFINITY (BUT NOT BEYOND!)
x ≥ 3
We often use Infinity in interval notation. Infinity is not a real number, in this case it
just means "continuing on ..."
For example, x greater than or equal to, 3:
[3, +∞)
Note that we use the round bracket with infinity, because we don't reach it!
There are 4 possible "infinite ends":
Interval
Inequality
(a, +∞)
x>a
"greater than a"
[a, +∞)
x≥a
"greater than or equal to a"
(-∞, a)
x<a
"less than a"
(-∞, a]
x≤a
"less than or equal to a"
We could even show no limits by using this notation: (-∞, +∞) =
ℝ
Exercise. Sketch the graph of the given interval:
a) (−3,2]
c) [−1,+∞)
b) (−∞,−5)
d) (0,6)
e) {x ∈ ℝ / −2 ≥ x}
f) {x ∈ ℝ / −1 < x < 4}
Exercise. Use both interval notation and inequalities to describe the interval shown on the graph:
a)
b)
c)
3.1. Integer part and fractional part
The integer part of a real number is the greatest integer which is less than or equal to that number.
The fractional part of a real number is the difference between then number and its integer part.
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3.2. Approximations
An approximation of a number is a representation of that number that is not exact, but still close
enough to be useful.
There are two important ways to approximate real numbers.
•
Truncation: to approximate a number to the tenths (units, tens, hundredths, thousandths…) by
truncation is to omit the figures at the right of the tenths.
For example, to truncate these numbers to the hundredths:
4.1798 ≈ 4.17
3,89176 ≈ 3.89
6,0092 ≈ 6
REMEMBER
•
≈
means "approximately equal to"
Rounding off: to round a number to the tenths (units, tens, hundredths, thousandths…) is to
find out an approximation by following this rule: you watch the figure at the number you want
to approximate and…
▪ If the following figure is less than five, you consider the following figures as zero. For
example, if you want to round off 3.45412 to the hundredths, as the following figure is
4 and it’s less than five, the following figures are considered as zero: 3.45 would be the
approximation. Truncation and rounding off are the same in this case.
▪ If the following figure is equal to or greater than than five, you add one (to the
figure in the position you want to approximate to). For example, to approximate
30752.652 to the tenths, as the following figure is 5 and it’s greater than or equal to
five, we have to change 6 by 7 and the rest of the following digits are considered as
zero: 30753.7 would be the approximation.
3.3. Absolute and percentage error
•
If a is an approximation of a real value v, the absolute error of the approximation is
e a =∣v−a∣
The error is expressed using the real value unit of measure.
For example, the real height of Peter is 1.5943 m, but he always says his height
is 1.60 m. This value, 1.60 m, is an approximation for the real value. As 1.60 m is
not the real length, there is an error of approximation.
The absolute error of the approximation can be calculted this way:
e a = │1.60 - 1.5943│ = │-0.0057│ = 0.0057 m.
•
The percentage error shows the error as a percent of the exact value. To calculate it you
just divide the absolute error by the exact value and make it a percentage:
ea
e r= ⋅100
v
For example, for Peter, we have this percentage error:
e r=
absolute error
real value
⋅100=
0.0057
1.6057
⋅100 = 0.354 %
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Exercise. Complete the chart:
Real value
Approximation
1h 5m 10s
1h 5m 7s
23.52 cm
1456.76 km
30º 5’
50.20501 g
50.30 g
Absolute error
Percentage error
0.04 cm
3.4. Standard form
Standard form is a way of writing down very large or very small numbers easily.
For example, 10³ = 1000, so 4 × 10³ = 4000. So 4000 can be written as 4 × 10³.
This idea can be used to write even larger numbers down easily in standard form.
Small numbers can also be written in standard form. However, instead of the exponent being positive
(in the above example, the exponent was 3), it will be negative.
The rules when writing a number in standard form is that first you write down a number between 1
and 10, and then you write × 10 to a power that puts the decimal point where it should be (i.e. it shows
how many places to move the decimal point).
In British English, "Standard Form" is just another name for Scientific Notation.
However, in other countries it just means "not in expanded form":
Standard Form
Expanded Form
561
500 + 60 + 1
It is used a lot in Science. For example, the Sun has a Mass of 1.988 × 10 30 kg.
It would be too hard for scientists to have to write 1,988,000,000,000,000,000,000,000,000,000 kg.
4.1. Factorial of a number
The factorial of a non-negative integer n, denoted by n!, is the product of all positive
integers less than or equal to n.
For example:
4! = 4 ∙ 3 ∙ 2 ∙ 1 = 24
7! = 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1 = 5,040
This is pronounced "4 factorial".
This is pronounced "7 factorial".
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There are two exceptions to this rule:
0! = 1
and
1! = 1
The factorial of a number is used to find out permutations, which are all possible arrangements of a
collection of things, where the order is important.
For example, you want to visit the homes of three friends Alex ("a"), Betty ("b") and Chandra ("c"), but
haven't decided in what order. What choices do you have?
You have a total of 6 choices: {a,b,c} {a,c,b} {b,a,c} {b,c,a} {c,a,b} {c,b,a}
You can find out the total number of choices by working out the factorial.
3! = 3 ∙ 2 ∙ 1 = 6
If the order does not matter, it is a combination.
4.2. Combinations
In Mathematics, a kcombination or a binomial coefficient is a number defined by this formula:
It is often called "n choose r" (such as "16 choose 3").
For example,
8!
8!
6⋅7⋅8
8 =
=
=
=56
3 3!⋅( 8−3) ! 3!⋅5! 1⋅2⋅3
()
Binomial coeeficients are used to find out the number of combinations without repetition (this is how
lotteries work!).
With combinations, it is important to assume that the order does not matter.
For example, if you are making a sandwich, how many different combinations of 2 ingredients could you
make with cheese, ham and turkey?
The aswer, of course, is three: {cheese, ham}, {cheese, turkey} or {ham, turkey}.
To find out the number of combinations, we can just calculate:
3!
3!
3
3 =
=
= =3
2!⋅
(3−2)!
2
!
⋅
1
!
1
2
()