Numbers, Sets, and
Operations
The words number and numeral are often used as if they mean the same thing.
But they’re different. A number is an abstraction. You can’t see or feel a number.
A numeral is a tangible object, or a group of objects, that represents a number.
Suppose you buy a loaf of bread cut into eighteen slices. You can consider the
whole sliced-up loaf as a numeral that represents the number eighteen, and each
slice as a digit in that numeral. You can’t eat the number eighteen, but you can eat
the bread.
Figuring with ngers
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Imagine it’s the afternoon of the twenty-fourth day of July. You have a doctor’s
appointment for the afternoon of the sixth of August. How many days away is
your appointment?
A calculator won’t work very well to solve this problem. Try it and see! You can’t
get the right answer by any straightforward arithmetic operation on twenty-four
and six. If you attack this problem as I would, you’ll count out loud starting with
tomorrow, July twenty- fth (under your breath): “twenty- ve, twenty-six, twentyseven, twenty-eight, twenty-nine, thirty, thirty-one, one, two, three, four, ve,
six!” While jabbering away, I would use my ngers to count along or make “hash
marks” on a piece of paper (Fig. 1-1). You might use a calendar and point to the
days one at a time as you count them out. However you do it, you’ll come up with
thirteen days if you get it right. But be careful! This sort of problem is easy to
mess up.
Don’t be embarrassed if you nd yourself guring out simple problems like this
using your ngers or other convenient objects. You’re making sure that you get
the right answer by using numerals to represent the numbers. Numerals are tailormade for solving number problems because they make abstract things easy to
envision.
In the Hindu-Arabic numeration system, large numbers are represented by
building up numerals digit-by-digit from right to left, giving each succeeding
digit ten times the value of the digit to its right. They gave each digit more or less
“weight” or value, depending on where it was written in relation to other digits in
the same numeral. The idea was that every digit in a numeral should have ten
times the value of the digit (if any) to its right.
0 (zero) is also called a “cipher"
The radix or base of a numeration system is the number of single-digit symbols it
has. The radix-ten system, also called base-ten or the decimal numeration system,
therefore has ten sym- bols, not counting commas (or decimal points
Natural Numbers
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Mathematicians de ne the number represented by the entire set N as a form of
“in nity” and denote it using the last letter in the Greek alphabet, omega, in
lowercase (ω). “Omega” is a traditional expression for “the end of all things.” In
formal terms, ω is called an in nite ordinal or trans nite ordinal, and it has some
strange properties.
Is there a largest prime?
Now that you know what a prime number is, and you know that any
nonprime natural num- ber can be broken down into a product of primes,
you might ask, “Is there a largest prime?” The answer is “No.” Here’s why.
You might have to read the following explanation two or three times to
completely understand it. Try to follow it step-by-step. If you can accept
each step of this argument one at a time, that’s good enough. The fact that
there is no such thing as a largest prime is one of the most important facts,
or theorems, that have ever been proven in mathematics.
Let’s start by imagining that there actually is a largest prime number. Then
we’ll prove that this assumption cannot be true by “painting ourselves into
a corner” where we end up with something ridiculous. Now that we have
decided there is a largest prime, suppose we give it a name. How about p?
Theoretically, we can list the entire set of prime numbers (call it P). It
might take mountains of paper and centuries of time, but if there is a largest
prime, we can eventually write all of the primes. We can describe the set P
in shorthand like this:
P = {2, 3, 5, 7, 11, 13, ..., p}
Suppose that we multiply all of these primes together. We get a composite
number, because it is a product of primes. No doubt, this number is huge—
larger than any calculator can
display—but it will be nite. Let’s call it y. What if we add 1 to y, getting a
number even larger than the product of all the primes? If you call that new
number z, you can express it like this:
z=y+1
= (2 × 3 × 5 × 7 × 11 × 13 × ... × p) + 1
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Now we know that z has to be larger than p, because z is 1 more than, say, 2 × p
or 3 × p or 5 × p or 7 × p. But there’s something else interesting about z. If we
divide z by any prime number, we always get a remainder of 1. That’s because if
we divide y by any prime, there’s no remainder, and z is exactly 1 more than y.
We know that z can’t be prime, because we’ve already determined that z is bigger
than p, and we have already assumed that p is the largest prime. So z is
composite. Because z is composite, it must be divisible without a remainder by at
least one prime, that is, one element of set P. But wait! We just gured out a
minute ago that if we divide z by any element of P, we get a remainder of 1.
Therefore, z can’t be composite. But it can’t be prime either. But every natural
number larger than 1 is either prime or composite! But ... but ... but ... we are
trapped!
There’s only one way out of this situation. Our original assumption, that there is a
largest prime number, must be false.
When the absolute value of the numerator in a fraction is larger than, or equal to,
the absolute value of the denominator, some people call it an improper fraction.
A positive-integer power is a quantity multiplied by itself a certain number of
times. If a non- zero quantity is divided by itself once, it is said to be “raised” to
the zeroth power, and the result is always 1. A negative-integer power is a
nonzero quantity divided by itself more than once. Powers are denoted by
exponents. Decimal notation is based on integer powers of 10. The number 10 is
called the exponential base. It can also be called simply the base or the radix.
The 0th power
By convention, anything raised to the 0th power is equal to 1. Anything except 0
itself, that is! The quantity 00 is not de ned.
x0 =1 iff x = a nonzero real number
The square root
If ap = b, then b1/p = a
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Higher roots
When p is a positive integer equal to 4 or more, people write or talk about the
numerical pow- ers and roots directly. That’s because geometric hypercubes
having 4 dimensions or more are not commonly named. A 4-dimensional
hypercube is technically called a tesseract, but you should expect incredulous
stares from your listeners if you say “2 tesseracted is 16” or “The tesseract root
of 81 is 3.”
N⊂Z⊂Q⊂R
Mathematicians use the symbol 0‫( א‬called aleph-null ) to describe the number of
elements (Cardinality of a set) in the set N of natural numbers.
Linear Equations and Relations
Types of mappings
Injection = one-to-one
Surjection = onto
Bijection = both
mapping
mapping
one-to-one and onto
slope- intercept (SI) form
y = mx + c
the constant m is the slope of the graph and the constant c is the y-intercept.
y−y0 =m(x−x0)
where x is the independent variable, y is the dependent variable, m is the slope,
and (x0, y0) are the coordinates of a known point on the graph.
Radian measure There is another unit for measurement of an angle, called the
radian measure. Angle subtended at the centre by an arc of length 1 unit in a unit
circle (circle of radius 1 unit) is said to have a measure of 1 radian.
if in a circle of radius r, an arc of length l subtends an angle θ radian at the centre,
we have
θ=
l
r
or l = r θ .
2π radian = 360° or π radian = 180°
1 radian =
180°/
π = 57° 16ʹ approximately.
Opposite-Angle Identities
sin(–θ) = –sin θ
cos(–θ) = cos θ
tan(–θ) = –tan θ
1° = π/
radian = 0.01746 radian approximately.
180
Radian measure = π/
Degree measure
180 ×
Degree measure =
180/
π × Radian measure
for all
real x,
sin2x+cos2x=1
1 + tan2 x
= sec2 x
1 + cot2 x
= cosec2 x
cos (– x)
= cos x
sin(–x)=–
sin x
–1≤cos x≤1and–
1≤sin x≤1 for all x
Pythagorean Identities
sin2 θ + cos2 θ = 1 or sin2 θ = 1 – cos2 θ or cos2 θ = 1 – sin2 θ tan2 θ + 1 = sec2 θ or tan2 θ =
sec2 θ – 1
1 + cot2 θ = csc2 θ or cot2 θ = csc2 θ – 1
Sum and Difference Identities
sin(α + β) = sin α cos β + cos α sin β
sin(α – β) = sin α cos β – cos α sin β
cos(α + β) = cos α cos β – sin α sin β
cos(α – β) = cos α cos β + sin α sin β
cos (x + y) = cos x cos y – sin x sin y
cos (x – y) = cos x cos y + sin x sin y
sin (x + y) = sin x cos y + cos x sin y
sin (x – y) = sin x cos y – cos x sin y
cos ( π/ – x ) = sin x
2
sin ( π/ – x ) = cos x
2
cos (π/ +x) = – sin x
2
sin (π/ +x) = cos x
2
cos(π–x) = – cos x
sin(π–x) = sin x
cos(π+x) = – cos x
sin(π+x) = – sin x
cos (2π – x) = cos x
sin (2π – x) = – sin x
If none of the angles x, y and (x+y) is an odd multiple of π/ ,
2
then tan (x + y) = (tan x + tan y) / (1 – tan x tan y)
tan (x - y) = (tan x - tan y) / (1 + tan x tan y)
If none of the angles x, y and (x + y) is a multiple of π, then
cot (x+y) = cot x cot y – 1 / (cot y + cot x)
cot (x-y) = cot x cot y + 1 / (cot y - cot x)
cos 2x = cos2x – sin2 x = 2 cos2 x – 1 = 1 – 2 sin2 x = 1–tan2 x /1+ tan2 x
≠
sin 2x = 2 sinx cos x = 2tan x / (1+ tan2 x), x
n π + π/2
tan2x= 2tan x / (1+ tan2 x) if 2x nπ+ π/2
sin 3x =3sinx–4sin3x
cos 3x=4cos3x–3cosx
3tanx–tan3 x π / 1–3tan2x if3x nπ+π/
tan3x=
2
sin2θ =2sinθcosθ
cos2θ =1−2sin2θ
tan2θ = 2tanθ / 1−tan θ
2
(i) 2cosxcosy=cos(x+y)+cos(x–y)
(ii) –2sinxsiny=cos(x+y)–cos(x–y)
≠
≠
(iii) 2sinxcosy=sin(x+y)+sin(x–y)
(iv) 2cosxsiny=sin(x+y)–sin(x–y).
sin x = 1/ “x is equal to the angle whose sine is equal to 1/ .
2
2
To uniquely determine a triangle ( nd only one possible shape and size), you need
✓ SSS: The measures of the three sides
✓ SAS: The measures of two sides and the angle between them
✓ ASA: The measures of two angles and the side between them
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✓ AAS: The measures of two angles and one of the sides
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Finding area with base and height
The equation for the area, A, of a triangle with base b and height h is A = 1 bh
2
Finding area with three sides
Heron’s formula
A2= s(s−a)(s−b)(s−c).
Finding area with SAS
If triangle ABC has sides measuring a, b, and c opposite the respective angles, you can
nd the area with one of these formulas:
A= 1/ absinC
2
A= 1/ bcsinA
2
A= 1/ acsinB
2
Finding area with ASA
2
Area = a sinBsinC / 2sinA
2
Area = b sinAsinC / 2sinB
2
Area = c sinAsinB / 2sinC
sin 3θ = 3 sin θ – 4 sin3 θ
cos 3θ = 4 cos3 θ – 3 cos θ
3
2
tan3θ = 3tanθ −tan θ /1−3tan θ
sin A cos B = 1 ⎡⎣ sin ( A + B ) + sin ( A − B )⎤⎦
2
sinAsinB= 1 ⎡⎣cos(A−B)−cos(A+B)⎤⎦
2
cos A sin B = 1 ⎡⎣ sin ( A + B ) − sin ( A − B )⎤⎦
2
c o s A c o s B = 1 ⎡⎣ c o s ( A − B ) + c o s ( A + B ) ⎤⎦
2
sin A + sin B = 2 sin ( A + B /2)cos ( A − B /2)
sin A − sin B = 2 cos ( A + B /2 )sin ( A − B /2 )
cos A + cos B = 2 cos ( A + B /2)cos ( A − B /2)
cos A − cos B = −2 sin ( A + B /2 )sin ( A − B /2 )
The plane having a complex number assigned to each of its point is called the
complex plane or the Argand plane.
Polar representation of a complex number
x = r cos θ, y = r sin θ and therefore, z = r (cos θ + i sin θ). The latter is said to be
the polar form of the complex number. Here r= x2 + y2 = |z| is the
modulus of z and θ is called the argument (or amplitude) of z which is denoted by
arg z.
A permutation is an arrangement in a de nite order of a number of objects taken
some or all at a time.
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Theorem 1 The number of permutations of n different objects taken r at a time,
where0<r≤n and the objects do not repeat is n(n–1)(n–2)...(n–r+1), which is
denoted by nPr.
Theorem 2 The number of permutations of n different objects taken r at a time,
where repetition is allowed, is nr.
Theorem 3 The number of permutations of n objects, where p objects are of the
n! same kind and rest are all different = n!/
p!
.
In fact, we have a more general theorem.
Theorem 4 The number of permutations of n objects, where p1 objects are of one
kind, p2 are of second kind, ..., pk are of kth kind and the rest, if any, are of different
kind is n! / p1! p2! ... pk!
Theorem5nP=nC r!,0<r≤n.
Theorem 6 nCr +n Cr−1 = n+1Cr
Binomial theorem for any positive integer n,
(a + b)n = nC0an + nC1an–1b + nC2an–2 b2 + ...+ nCn – 1a.bn–1 + nCnbn
Area of a triangle
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Heron’s Formula reads: A2 = s ( s − a )( s − b )( s − c ) where a, b, and c are
the lengths of the sides of the triangle and s is the sem perimeter (half the perimeter).