Number Theory and the Real Number System Theodore Vassiliadis
Dec 31, 2015
Number Theory and the Real Number System
Theodore Vassiliadis
An introduction to number theory• Prime numbers• Integers, rational numbers, irrational
numbers, and real numbers• Properties of real numbers• Rules of exponents and scientific notation
WHAT YOU WILL LEARN
Number TheoryThe study of numbers and their properties.The numbers we use to count are called
natural numbers, , or counting numbers.
{1,2,3, 4,5,...}
FactorsThe natural numbers that are multiplied
together to equal another natural number are called factors of the product.
Example: The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and
24.
DivisorsIf a and b are natural numbers and the
quotient of b divided by a has a remainder of 0, then we say that a is a divisor of b or a divides b.
Prime and Composite NumbersA prime number is a natural number greater
than 1 that has exactly two factors (or divisors), itself and 1.
A composite number is a natural number that is divisible by a number other than itself and 1.
The number 1 is neither prime nor composite, it is called a unit.
Rules of Divisibility
285The number ends in 0 or 5.5
844 since 44 4
The number formed by the last two digits of the number is divisible by 4.
4
846 since 8 + 4 + 6 = 18
The sum of the digits of the number is divisible by 3.
3
846The number is even.2
ExampleTestDivisible by
Divisibility Rules, continued
730The number ends in 0.10
846 since 8 + 4 + 6 = 18
The sum of the digits of the number is divisible by 9.
9
3848since 848 8
The number formed by the last three digits of the number is divisible by 8.
8
846The number is divisible by both 2 and 3.
6
ExampleTestDivisible by
The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a unique product of prime numbers.
This unique product is referred to as the prime factorization of the number.
Division Method1. Divide the given number by the smallest
prime number by which it is divisible.2. Place the quotient under the given number.3. Divide the quotient by the smallest prime
number by which it is divisible and again record the quotient.
4. Repeat this process until the quotient is a prime number.
Example of division methodWrite the prime factorization of 663.
The final quotient 17, is a prime number, so we stop. The prime factorization of 663 is 3 •13 •17
13
3
17
221
663
Finding the LCM of Two or More Numbers Determine the prime factorization of each
number. List each prime factor with the greatest
exponent that appears in any of the prime factorizations.
Determine the product of the factors found in step 2.
Example (LCM)Find the LCM of 63 and 105.
63 = 32 • 7105 = 3 • 5 • 7
Greatest exponent of each factor:32, 5 and 7
So, the LCM is 32 • 5 • 7 = 315.
Whole NumbersThe set of whole numbers contains the set of
natural numbers and the number 0.Whole numbers = {0,1,2,3,4,…}
IntegersThe set of integers consists of 0, the natural
numbers, and the negative natural numbers. Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…}On a number line, the positive numbers
extend to the right from zero; the negative numbers extend to the left from zero.
Writing an InequalityInsert either > or < in the box between the paired numbers to make the statement correct.
a) 3 1 b) 9 7 3 < 1 9 < 7c) 0 4 d) 6 8 0 > 4 6 < 8
The Rational Numbers• The set of rational numbers, denoted by
Q, is the set of all numbers of the form p/q, where p and q are integers and q 0.
• The following are examples of rational numbers:
1
3,
3
4,
7
8, 1
2
3, 2, 0,
15
7
FractionsFractions are numbers such as:
The numerator is the number above the fraction line.
The denominator is the number below the fraction line.
1
3,
2
9, and
9
53.
Reducing FractionsIn order to reduce a fraction to its lowest
terms, we divide both the numerator and denominator by the greatest common divisor.
Example: Reduce to its lowest terms.
Solution:
72
81
72 72 9 8
81 81 9 9
Terminating or Repeating Decimal NumbersEvery rational number when expressed as
a decimal number will be either a terminating or a repeating decimal number.
Examples of terminating decimal numbers are 0.7, 2.85, 0.000045
Examples of repeating decimal numbers 0.44444… which may be written
0.4,
and 0.2323232323... which may be written 0.23.
Multiplication of Fractions
Division of Fractions
a
b
c
d
a c
b d
ac
bd, b 0, d 0
a
b
c
d
a
b
d
c
ad
bc, b 0, d 0, c 0
Example: Multiplying FractionsEvaluate the
following.
a)
b)
2
3
7
16
2
3
7
16
27316
14
48
7
24
1
3
4
2
1
2
13
4
2
1
2
7
45
2
35
84
3
8
Example: Dividing FractionsEvaluate the
following.a)
b)
2
3
6
7
2
3
6
7
2
37
6
2736
14
18
7
9
5
8
4
5
5
8
4
5
5
85
4
5584
25
32
Addition and Subtraction of Fractions
a
c
b
c
a b
c, c 0;
a
c
b
c
a b
c, c 0
Example: Add or Subtract Fractions
Add:
Subtract:
4
9
3
9
4
9
3
9
4 3
9
7
9
11
16
3
16
11
16
3
16
11 3
16
8
16
1
2
Fundamental Law of Rational NumbersIf a, b, and c are integers, with b 0, c 0,
then
a
b
a
bc
c
acbc
ac
bc.
Example:Evaluate:
Solution:
7
12
1
10.
7
12
1
10
7
125
5
1
106
6
35
60
6
60
29
60
Irrational NumbersAn irrational number is a real number
whose decimal representation is a nonterminating, nonrepeating decimal number.
Examples of irrational numbers: 5.12639573...
6.1011011101111...
0.525225222...
Radicals are all irrational
numbers. The symbol is called the radical sign. The number or expression inside the radical sign is called the radicand.
2, 17, 53
Perfect SquareAny number that is the square of a natural
number is said to be a perfect square.The numbers 1, 4, 9, 16, 25, 36, and 49 are
the first few perfect squares.
Real NumbersThe set of real numbers is formed by the
union of the rational and irrational numbers.The symbol for the set of real numbers is .
Relationships Among Sets
Irrational numbers
Rational numbers
Integers
Whole numbersNatural numbers
Real numbers
Properties of the Real Number System Closure
If an operation is performed on any two elements of a set and the result is an element of the set, we say that the set is closed under that given operation.
Commutative PropertyAddition
a + b = b + a for any real numbers a and b.
Multiplication a • b = b • a for any real numbers a and b.
Example8 + 12 = 12 + 8 is a true statement.5 9 = 9 5 is a true statement.
Note: The commutative property does not hold true for subtraction or division.
Associative PropertyAddition (a + b) + c = a + (b +
c),
for any real numbers a, b, and c.
Multiplication (a • b) • c = a • (b • c),
for any real numbers a, b, and c.
Example(3 + 5) + 6 = 3 + (5 + 6) is true.
(4 6) 2 = 4 (6 2) is true.
Note: The associative property does not hold true for subtraction or division.
Distributive PropertyDistributive property of multiplication over
addition
a • (b + c) = a • b + a • c
for any real numbers a, b, and c.
Example: 6 • (r + 12) = 6 • r + 6 • 12 = 6r + 72
ExponentsWhen a number is written with an exponent,
there are two parts to the expression: baseexponent
The exponent tells how many times the base should be multiplied together.
45 44444
Scientific NotationMany scientific problems deal with very large
or very small numbers.93,000,000,000,000 is a very large number.0.000000000482 is a very small number.
Scientific Notation continuedScientific notation is a shorthand method
used to write these numbers.
9.3 1013 and 4.82 10–10 are two examples of numbers in scientific notation.
To Write a Number in Scientific Notation1. Move the decimal point in the original number to
the right or left until you obtain a number greater than or equal to 1 and less than 10.
2. Count the number of places you have moved the decimal point to obtain the number in step 1.If the decimal point was moved to the left, the count is to be considered positive. If the decimal point was moved to the right, the count is to be considered negative.
3. Multiply the number obtained in step 1 by 10 raised to the count found in step 2. (The count found in step 2 is the exponent on the base 10.)
ExampleWrite each number in scientific notation.
a) 1,265,000,000.1.265 109
b) 0.0000000004324.32 1010
To Change a Number in Scientific Notation to Decimal Notation1. Observe the exponent on the 10.2. a) If the exponent is positive, move the
decimal point in the number to the right the same number of places as the exponent. Adding zeros to the number might be necessary.
b) If the exponent is negative, move the decimal point in the number to the left the same number of places as the exponent. Adding zeros might be necessary.
ExampleWrite each number in decimal notation.
a) 4.67 105
467,000
b) 1.45 10–7
0.000000145