This document provides an overview of basic number theory concepts including different number systems, set notation, order of operations, rational numbers, real numbers, properties of real numbers, and finite and repeating decimals. It defines natural numbers, integers, rational numbers, and real numbers. It covers identifying subsets of numbers, rational number operations, set notation, order of operations, properties such as closure and identities, representing decimals as fractions, and converting between decimals and fractions.

1. 0.1 Number Theory
Unit 0 Review of Basic Concepts

2. Concepts and Objectives
⚫ Number Theory
⚫ Identify subsets of numbers
⚫ Simplify expressions using order of operations
⚫ Identify real number axioms
⚫ Rational number operations

3. Number Systems
⚫ What we currently know as the set of real numbers was
only formulated around 1879. We usually present this
as sets of numbers.

4. Number Systems
⚫ The set of natural numbers () and the set of integers
() have been around since ancient times, probably
prompted by the need to maintain trade accounts.
Ancient civilizations, such as the Babylonians, also used
ratios to compare quantities.
⚫ One of the greatest mathematical advances was the
introduction of the number 0.

5. Set Notation
⚫ The basic definition of a set is a collection of objects, and
the objects that belong to that set are called the
elements or members of that set.
⚫ We use braces, { }, to list the elements in a set, for
example, {1, 2, 3, 4}. To show that 4 is a member of the
set, we use the symbol, , and write 4  {1, 2, 3, 4}.
⚫ We use ellipsis (…) to show that a set continues
according to an established pattern. Ex.: {1, 2, 3, …, 10}
⚫ {x | x   and 2 < x < 7} would be read “the set of all
elements x such that x is a natural number between 2
and 7.”

6. Set Notation (cont.)
⚫ If a set has no elements, it is referred to the null set or
empty set, and it is denoted as either { } or . For
example, the set of all odd numbers ending in 2 would be
{ }.
⚫ If every member of set A is a member of set B, then set A
is a subset of set B, which is written A  B.
⚫ Given two sets A and B, the set of all elements belonging
to both set A and set B is called the intersection of the
two sets, written A  B. The set of all elements belong to
either set A or set B is called the union of the two sets,
written A  B.

7. Order of Operations
⚫ Parentheses (or other grouping symbols, such as square
brackets or fraction bars) – start with the innermost set,
following the sequence below, and work outward.
⚫ Exponents
⚫ Multiplication
⚫ Division
⚫ Addition
⚫ Subtraction
working from left to right
working from left to right

8. Order of Operations
⚫ Use order of operations to explain why
( )−  −
2 2
3 3

9. Order of Operations
⚫ Use order of operations to explain why
⚫ We can think of –3 as being –1  3. Therefore we have
It should be easier now to see that on the left side we
multiply first and then apply the exponent, and on the
right side, we apply the exponent and then multiply.
( )−  −
2 2
3 3
( )−  −
2 2
1 3 1 3

10. Order of Operations
Work the following examples without using your calculator.
1.
2.
3.
− + 2 5 12 3
( ) ( )( )− − + −
3
4 9 8 7 2
( )( )
( )
8 4 6 12
4 3
− + − − 
− −

11. Order of Operations
Work the following examples without using your calculator.
1.
2.
3.
− + 2 5 12 3
( ) ( )( )− − + −
3
4 9 8 7 2
( )( )
( )
− + − − 
− −
8 4 6 12
4 3
1. –6
2. –60
−
6
3.
7

12. Rational Numbers
⚫ A number is a rational number () if and only if it can be
expressed as the ratio (or Quotient) of two integers.
⚫ Rational numbers include decimals as well as fractions.
The definition does not require that a rational number
must be written as a quotient of two integers, only that it
can be.

13. Examples
⚫ Example: Prove that the following numbers are
rational numbers by expressing them as ratios of
integers.
1. 2-4 4.
2. 64-½ 5.
3. 6. –5.4322986


4
20.3
0.9
6.3

14. Examples
⚫ Example: Prove that the following numbers are
rational numbers by expressing them as ratios of
integers.
1. 2-4 4.
2. 64-½ 5.
3. 6. –5.4322986


4
20.3
0.9
6.3
1
16
1
8
4
1
7
=
1 61
20
3 3
−
54322986
10000000

15. Real Numbers
⚫ The number line is a geometric model of the system of
real numbers. Rational numbers are thus fairly easy to
represent:
⚫ What about irrational numbers? Consider the following:
•
(1,1)
2
This comes from the
Pyth. Theorem.
12 + 12 = c2 , so
2c =

16. Real Numbers
⚫ In this way, if an irrational number can be identified
with a length, we can find a point on the number line
corresponding to it.
⚫ What this emphasizes is that the number line is
continuous—there are no gaps.

17. Properties of Real Numbers
⚫ Closure Property
⚫ a + b  
⚫ ab  
⚫ Commutative Property
⚫ a + b = b + a
⚫ ab = ba
⚫ Associative Property
⚫ (a + b) + c = a + (b + c)
⚫ (ab)c = a(bc)
⚫ Identity Property
⚫ a + 0 = a
⚫ a  1 = a
⚫ Inverse Property
⚫ a + (–a) = 0
⚫
⚫ Distributive Property
⚫ a(b ± c) = ab ± ac
For all real numbers a, b, and c:
1
=1a
a

18. Properties of Real Numbers
⚫ The properties are also called axioms.
⚫ 0 is called the additive identity and 1 is called the
multiplicative identity.
⚫ Notice the relationships between the identities and the
inverses (called the additive inverse and the
multiplicative inverse).
⚫ Saying that a set is “closed” under an operation (such as
multiplication) means that performing that operation on
numbers in the set will always produce an answer that is
also in the set – there are no answers outside the set.

19. Properties of Real Numbers
⚫ Examples
⚫ The set of natural numbers () is not closed under the
operation of subtraction. Why?
⚫ –20  5 = –4. Does this show that the set of integers is
closed under division?

20. Properties of Real Numbers
⚫ Examples
⚫ The set of natural numbers () is not closed under the
operation of subtraction. Why?
⚫ You can end up with a result that is not a natural
number. For example, 5 – 7 = –2, which is not in .
⚫ –20  5 = –4. Does this show that the set of integers is
closed under division?
⚫ No. Any division that has a remainder is not in .

21. Finite and Repeating Decimals
⚫ If the decimal representation of a rational number has a
finite number of digits after the decimal, then it is said to
terminate.
⚫ If the decimal representation of a rational number does
not terminate, then the decimal is periodic (or
repeating). The repeating string of numbers is called the
period of the decimal.
⚫ It turns out that for a rational number where b > 0,
the period is at most b – 1.
a
b

22. Finite and Repeating Decimals
⚫ Example: Use long division to find the decimal
representation of and find its period.
What is the period of this decimal?
462
13
462
35.538461
13
=
6

23. Finite and Repeating Decimals
⚫ The repeating portion of a decimal does not necessarily
start right after the decimal point. A decimal which
starts repeating after the decimal point is called a
simple-periodic decimal; one which starts later is called a
delayed-periodic decimal.
1 2 30. ... td d d d ( 0)td
0.3, 0.142857, 0.09, 0.076923 1 2 30. ... pd d d d
0.16, 0.083, 0.0714285, 0.06
Type of Decimal Examples General Form
terminating 0.5, 0.25, 0.2, 0.125, 0.0625
simple-periodic
delayed-periodic + + + +1 2 3 1 2 30. ... ...t t t t t pd d d d d d d d

24. Decimal Representation
⚫ If we know the fraction, it’s fairly straightforward
(although sometimes tedious) to find its decimal
representation. What about going the other direction?
How do we find the fraction from the decimal, especially
if it repeats?
⚫ We can state that the terminating decimal 0.d1d2d3…dt
can be written as .
Example:
1 2 3...
10
t
t
d d d d
= =
845 169
0.845
1000 200

25. Decimal Representation
⚫ For simple-periodic decimals, the “trick” is to turn them
into fractions with the same number of 9s in the
denominator as there are repeating digits and simplify:
To put it more generally, to convert , we can
write it as .
3 1
0.3
9 3
= = = =
6 2
0.06
99 33
153846 2
0.153846
999999 13
= =
1 2 30. ... pd d d d
1 2 3...
999...9
pd d d d

26. Decimal Representation
⚫ For delayed-periodic decimals, the process is a little
more complicated. Consider the following:
What is the decimal representation of ?
is a product of what two fractions?
Notice that the decimal representation has
characteristics of each factor (2 terminating digits and 1
repeating digit).
1
12
0.083
1
12
1 1
4 3

27. Decimal Representation
⚫ It turns out you can break a delayed-periodic decimal
into a product of terminating and simple-periodic
decimals, so the general form is also a product of the
general forms:
The decimal can be written as the
fraction , where N is the integer
d1d2…dtdt+1dt+2…dt+p – d1d2…dt . (whew!)
+ + +21 2 10. ... ... pt t t td d d d d d
( )−10 10 1pt
N

28. Decimal Representation
⚫ Example: Convert the decimal to a fraction.
Comparing this to the formula, we can see that t = 1 (the
terminating part) and p = 3 (the repeating part).
Thus:
Another way to look at the denominator is that it is 3
nines (repeating) followed by 1 zero (terminating).
4.9331
( )1 3
5
49
4
49331 49282 24641
331
9990 49910 10
.9
1
−
= = =
−

29. Decimal Representation
⚫ Example: Convert the decimal to a
fraction.
0.467988654

30. Decimal Representation
⚫ Example: Convert the decimal to a
fraction.
It’s possible this might reduce, but we can see that there
are no obvious common factors (2, 3, 4, 5, 6, 8, 9, or 10),
so it’s okay to leave it like this.
0.467988654
−
= =
988654
99 0
467
467
00
0
467 467988187
. 9
9
88654
999909 9 99009

31. Classwork
⚫ Classwork: 0.1 WS
⚫ If you aren’t able to finish before you leave, it will be due
at the beginning of our next class.