Numbers
1. Natural numbers
Natural numbers are the counting numbers we use in everyday life.
0, 1, 2, 3, . . .
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
It is assumed that you know how to apply basic operations (+, −, ×,
÷ and exponents/powers) with natural numbers.
In Math 132, we don’t use strict bedmas. Brackets are always evaluated
before other operations, exponents (or powers) are next. Then we have
equal priority for division and multiplication. If they both appear, we
apply the earlier (leftmost) operation first. After all those operations
are done, we apply addition and subtraction with equal priority. For
example, in the formula 2 − 3 + 5, we’ll evaluate this as 2 − 3 + 5 =
−1 + 5 = 4, rather than the strict bedmas priority which would result
in −6.
It should be clear that you can combine natural numbers using addition
and multiplication, and the result will be a natural numbers. We say
that the natural numbers are closed under addition and multiplication.
As subtraction sometimes takes us outside the natural numbers (what
is 4 − 7?), we need to extend our numbers to allow subtraction.
1.1. Properties. Standard properties include commutativity (order doesn’t matter), associativity (brackets don’t matter) and distributivity (operations play nice with one another). We’ll look at all of these
in class, including when they don’t work.
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NUMBERS
1.2. Multiplication. One view of multiplication leads us to division. We have multiplied by some value (say doubled) and we wish to
reverse that multiplication. We do this by dividing by that value (say
divide by two). This leads to fractions, which ties together division
and multiplication. It also leads to proportions.
Another view, decomposing, ask us to write a number as a product of
smaller values. This is called factorisation, and each multiplier is called
a factor. Ignoring one, which is always a factor, we are interested in
values that cannot be factorised. These are called primes e.g. 2, 3, 5,
7 and 11 are the first five primes.
1.3. Comparing. Equality is the simplest measure to compare
numbers, either they are equal or they aren’t. These can be the result
of operations e.g. 2+5 = 7. Along with operations to combine numbers,
we’ll also look at comparing numbers. The standard inequalities are <
(less than e.g. 2 < 3), ≤ (less than or equal to e.g. 4 ≤ 4 and 4 ≤ 5),
> (greater than e.g. 6 > 4) and ≥ (greater than or equal to e.g. 6 ≥ 6
and 7 ≥ 6).
1.4. Powers. Powers represent repeated multiplication, so 25 =
2 × 2 × 2 × 2 × 2 and 52 = 5 × 5 and 31 = 3. By examining how
multiplication and division work, we can see that xa × xb = xa+b ,
xa ÷ xb = xa−b and (xa )b = xa×b . This leads to the convention x0 = 1
e.g. 30 = 1.
1.5. Base ten. Our number system is called base 10, as every
digit represents a power of 10 e.g. 249 = 2 × 102 + 4 × 10 + 9. This
allows us to represent arbitrarily large numbers, and later we’ll extend
this to represent arbitrarily small numbers as well.
2. Integers
Integers are an extension of the natural numbers, they include negative
numbers as well. Each natural number has a negative value associated
with it e.g. − 3 is negative three. For Math 132, we’ll consider them as
the minimum requirement to allow us to do subtraction freely. Note
the following rules, which will come in handy later
a +− b = a − b and a −− b = a + b
3. RATIONALS
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We still want to be able to do multiplication, and we have the following
rules
a ×− b =− (a × b) and − a ×− b = a × b
In this sense, each negative number can be considered to be −1 times
the associated positive number e.g. −4 = −1 × 4 and −6 = −1 × 6.
The integers are closed under addition, multiplication and subtraction.
The next number system will allow us to do division.
2.1. Inequalities. Inequalities for integers are also a little more
difficult than for natural numbers. The following two rules apply.
(1) Negative numbers are always less than positive numbers e.g.
−8 < 5.
(2) When comparing two negative numbers, the inequality is the
reverse of the situation if they were both positive e.g. −4 <
−3.
One way to understand these rules is to describe it in a different way.
One number x is greater than another y whenever their difference x − y
is strictly positive (> 0). Using the above examples, 5 −− 8 = 13 which
is strictly positive, so 5 > −8. Similarly, −3 −− 4 = 1 which is strictly
positive, so −3 > −4. A good description will also work conversely e.g.
−8 − 5 = −13 which is not strictly positive so −8 is not greater than 5
(sometimes written as −8 ≯ 5, but we’d say −8 < 5).
You can see that multiplying through be a negative number reverses
the inequality e.g. −3 < 2 while −3 × −4 = 12 > −8 = −2 × 4.
Try to explain this using the difference description e.g 2 −− 3 > 0 so
(−4 × 2) − (−4 × −3) = −4 × (2 −− 3) = −4 × 5. The initial difference
was positive, but the final difference is negative.
3. Rationals
Rational numbers are sometimes called fractions. A fraction is represented by two numbers seperated by a line e.g. 2/3 or 23 .
A fraction can be represented by different pairs of numbers e.g
There is a rule for this, ab = dc whenever ad = bc.
3
4
= 68 .
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NUMBERS
Rationals allow us to do division freely ....... well ...... almost, as we
can never divide by 0. Remember we cannot divide by 0.
Here are some rules for combining fractions
a c
ad + bc a c
ac
a c
ad
+ =
, × =
and ÷ =
b d
bd
b d
bd
b d
bc
With the rules for multiplying and dividing by negative numbers we
see that a negative fraction can be represented with the sign in three
2
places e.g. − 25 = −2
= −5
(note we rarely write the denominator as
5
negative).
Note the similarity between the rules for multiplying and dividing fractions. The reciprocal or inverse of a fraction ab comes from flipping the
fraction, to ab e.g the reciprocal of 35 is 53 . Try to describe division using
a reciprocal and multiplication.
Inequalities for fractions are not too bad. To compare, you can make
both fractions have a positive common denominator, then compare the
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numerators e.g. 23 < 34 as 12
< 12
because 8 < 9. In a similar situation
to negatives, flipping a same signed fractions reverses the inequality
e.g. 32 < 43 but 32 > 34 . Can you use the difference description to make
sense of this (easier when both are positive fractions).
4. Reals
Real numbers are where intuition starts to falter a little. For us, real
numbers are decimal numbers (those with a finite number of digits
before the decimal point, but possibly infinitely many digits after).
The position immediately to the left of the decimal place is the ones (or
units). Every move to the left multiplies the digit by 10 and each move
to the right divides by 10 e.g. 249.35 is 9 units. 4 tens, 2 hundreds, 3
tenths and 5 hundreths. That is
249.35 = 2 × 100 + 4 × 10 + 9 × 1 + 3 ×
1
1
+5×
10
100
We don’t have time in the lectures to really get to grips with real
numbers, they are tricky beasts. A couple of topics that help are listed
below, you’ll see long division explained in the tutorials.
6. BIG IDEA
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Long division. This method of division lets us move from finite decimals to
infinite ones.
Sequences and series. A way to analyse patterns of numbers, especially useful for
decimals.
5. Comparing number systems
One way to compare the systems is to look at them extending the
numbers and operations.
limits, ÷, −, +, ×
÷, −, +, ×
−, +, ×
R
Q
Z
N
+, ×
0, 1, 2, . . .
−1, −2, . . .
6. Big Idea
At the end of each topic, we’ll discuss a big idea.
Take a natural (counting) number. Either you can write it as the
product of two smaller natural numbers (called factors) or you can’t
(called a prime number) e.g. 28 = 4 × 7, but 11 cannot be broken
down.
It is intuitive that you can break down a number into prime factors
(called a prime factorisation). What isn’t quite so intuitive is that
there is really only one prime factorisation for any number.
a
b
decimals
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NUMBERS
6.1. The Fundamental Theorem of Arithmetic. Any natural
number, greater than one, can be written as the product of primes, in
essentially one way.
For example, the number 40 can be written many ways (2)(2)(2)(5),
(2)(2)(5)(2), (2)(5)(2)(2), (5)(2)(2)(2), 23 × 5 and others. All these
essentially say the same thing, 40 is made up of three 2’s and one 5.
We’ll discuss why one part of this theorem (that it breaks down to
primes) is believable, but the other part isn’t so obvious (that there is
only one outcome).