General Mathematics - Number Systems and Sets

constructing a parallelogram

Addition of two complex numbers can be done geometrically by

Prime Factor

If a factor of a number is prime, it is called a

1. The associative laws of addition and multiplication. 2. The commutative laws of addition and multiplication. 3. The distributive law.

LAWS FOR COMBINING NUMBERS Numbers are combined in accordance with the following basic laws:

Members of Elements of the Set

The objects or symbols in a set are called

BASE of the number system

The place value which corresponds to a given position in a number is determined by the

a complex number is real if and only if it equals its conjugate.

The real and imaginary parts of a complex number can be extracted using the conjugate:

a curve, a surface or some other such object in n-dimensional space

an equation, or system of equations, in two or more variables defines

To separate a number into prime factors

begin by taking out the smallest factor If the number is even, take out all the 2's first, then try 3 as a factor

consecutive whole numbers

2,3,4,5,6

right-hand digit is even

A number is divisible by 2 if its

the sum of its digits is divisible by 3

A number is divisible by 3 if

the number formed by the three right-hand digits is divisible by 8

A number is divisible by 8 if

the sum of its digits is divisible by 9

A number is divisible by 9 if

Composite Number

A number that has factors other than itself and 1 is a

Absolute value and argument

Another way of encoding points in the complex plane other than using the x- and y-coordinates is to use the distance of a point P to O, the point whose coordinates are (0, 0) (the origin), and the angle of the line through P and O. This idea leads to the polar form of complex numbers.

The real part c and the imaginary part d of the denominator must not both be zero for division to be defined.

As shown earlier, c − di is the complex conjugate of the denominator c + di.

order of operations

G, E, M, A Grouping, Exponents, Multiply/Divide, Add/Subtract

Second Axiom of Equality

If the same quantity is subtracted from each of two equal quantities, the resulting quantities are equal. If equals are subtracted from equals, the results are equal.

The absolute value (or modulus or magnitude) of a complex number z = x + yi is

If z is a real number (i.e., y = 0), then r = |x|. In general, by Pythagoras' theorem, r is the distance of the point P representing the complex number z to the origin.

The multiplication of two complex numbers is defined by the following formula:

In particular, the square of the imaginary unit is −1: The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms.

positive

In the Rectangular Coordinate System, On the vertical line, direction upward is

negative

In the Rectangular Coordinate System, the direction to the left along the horizontal line is

positive

In the Rectangular Coordinate System, the direction to the right along the horizontal line is

7

No short method has been found for determining whether a number is divisible by

SUBSET relationship; T is said to be a subset of S.

Notice that every element of T is also an element of S. This establishes the

"reflection" of z about the real axis. In particular, conjugating twice gives the original complex number: .

The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted or . Geometrically, is the

the genus of the curve

The finiteness or not of the number of rational or integer points on an algebraic curve—that is, rational or integer solutions to an equation f(x,y) = 0, where f is a polynomial in two variables—turns out to depend crucially on

Place Value Concept

The number of digits in an integer indicates its rank; that is, whether it is "in the hundreds," "in the thousands," etc. The idea of rankong numbers in terms of tens, hundreds, thousands, etc., is based on the

Natural Numbers

The numbers which are used for counting in our number system are sometimes called

C or

The set of all complex numbers is denoted by

Here is called the modulus of a + bi, and the square root with non-negative real part is called the principal square root.

The square roots of a + bi (with b ≠ 0) are , where and where sgn is the signum function. This can be seen by squaring to obtain a + bi.

rectangular coordinates

This formula can be used to compute the multiplicative inverse of a complex number if it is given in

Inversive geometry

a branch of geometry studying more general reflections than ones about a line, can also be expressed in terms of complex numbers.

an equation in two variables defines

a curve in the plane

Positional notation (place value)

a form of coding in which the value of each digit of a number depends upon its position in relation to the other digits of the number. The convention used in our number system is that each digit has a higher place value than those digits to the right of it.

variable

a letter tat represents a number that is unknown (usually X or Y)

subtraction

decreased by

equation

has an equal sign (3x+5 = 14)

coefficient

the number touching the variable (in the case of 5x, would be 5)

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation:

the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.

Third Axiom of Equality

If two equal quantities are multiplied by the same quantity, the resulting products are equal. If equals are multiplied by equals, the products are equal.

which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra,

the number formed by the two right-hand digits is divisible by 4

A number is divisible by 4 if

righthand digit is 0 or 5

A number is divisible by 5 if its

even and the sum of its digits is divisible by 3

A number is divisible by 6 if it is

Prime Number

A number that has no factors except itself and 1 is a

Factor of the given number

Any number that can be divided lnto a given number without a remainder is a

Multiple of the given number

Any number that is exactly divisible by a given number is a

Odd Number

Any number that is not a multiple of 2 is an

Even Number

Any number that la a multiple of 2 is an

If the same quantity is added to each of two equal quantities, the resulting quantities are equal. If equals are added to equals, the results are equal.

First axiom of equality

Forth Axiom of Equality

If two equal quantities are divided by the same quantity, the resulting quotients are equal. If equals are divided by equals, the results are equal.

negative

In the Rectangular Coordinate System, On the vertical line, direction Downward is

s • {1, 2, 3, 4, 5, 6, 7, 8, 9}

S represent the set of all integers greater than 0 and less than 10. In symbols, this relationship could be stated as follows:

Letter

Since it is inconvenient to enumerate all of the elements of a set each time the set is mentioned, sets are often designated by a

Equal

Since the elements of the set {2, 4, e} are the same as the elements of{4, 2, e}, these two sets are said to be

T • {2, 4, 6, 8}

T, which comprises all positive even integers less than 10. This set is then defined as follows:

Digits

The Arabic numerals from 0 through 9 are called

base-ten number

The base which is most commonly used is ten, and the system with ten as a base is called the decimal system (decem is the Latin word for ten). Any number is assumed, unless indicated, to be a

solutions

The central problem of Diophantine geometry is to determine when a Diophantine equation has

magnitude and direction

The defining characteristic of a position vector is that it has

Numerals, Lines, or Points

The elements of a mathematical set are usually symbols, such as

1. If the same quantity is added to each of two equal quantities, the resulting quantities are equal. If equals are added to equals, the results are equal. 2. If the same quantity is subtracted from each of two equal quantities, the resulting quantities are equal. If equals are subtracted from equals, the results are equal. 3. If two equal quantities are multiplied by the same quantity, the resulting products are equal. If equals are multiplied by equals, the products are equal. 4. If two equal quantities are divided by the same quantity, the resulting quotients are equal. If equals are divided by equals, the results are equal.

The four axioms of equality in arithmetic and algebra are stated as follows: Note: These axioms are especially useful when letters are used to represent numbers.

one characteristic in common such as similarity of appearance or purpose

The objects in a set have at least

addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis).

These are emphasised in a complex number's polar form and it turns out notably that the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors:

Associative Law of Addition

This law can be applied to subtraction by changing signs in such a way that all negative signs are treated as number signs rather than operational signs.That is, some of the addends can be negative numbers.

Commutative Law of Addition

This law can be applied to subtraction by changing signs so that all negative signs become number signs and all signs of operation are positive.

Distributive Law

This law combines the operations of addition and multiplication. The distribution of a common multiplier among the terms of an additive expression.

Associative Law of Multiplication

This law states that the product of three or more factors is the same regardless of the manner in which they are grouped. Negative signs require no special treatment in the application of this law.

Commutative Law of Multiplication

This law states that the product of two or more factors is the same regardless of the order in which the factors are arranged. Negative signs require no special treatment in the application of this law.

Associative Law of Addition

This law states that the sum of three or more addends is the same regardless of the manner in which they are grouped. suggests association or grouping.

Commutative Law of Addition

This law states that the sum of two or more addends is the same regardless of the order in which they are arranged. Means to change, substitute or move from place to place.

counterclockwise through 90° about the origin: (a + bi)i = ai + bi2 = − b + ai.

Viewed in this way the multiplication of a complex number by i corresponds to rotating a complex number

Complex numbers

allow for solutions to certain equations that have no real solution: the equation has no real solution, since the square of a real number is 0 or positive.

Definition of genus

allow the variables in f(x,y) = 0 to be complex numbers; then f(x,y) = 0 defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, i.e., four dimensions). Count the number of (doughnut) holes in the surface; call this number the genus of f(x,y) = 0. Other geometrical notions turn out to be just as crucial.

repeated elements

are not necessary. That is, the elements of {2, 2, 3, 4} are simply {2, 3, and 4}

Number fields

are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K. (For example, the complex numbers C are an extension of the reals R, and the reals R are an extension of the rationals Q.)

Braces

are used to indicate sets

The numbers are conventionally plotted using the real part

as the horizontal component, and imaginary part as vertical These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.

quadratic field

consists of all numbers of the form , where a and b are rational numbers and d is a fixed rational number whose square root is not rational.

expression

does not have an equal sign (3x+5) (2a+9b)

Set

implies a collection or grouping of similar, objects or symbols.

Analytic number theory

in terms of its tools, as the study of the integers by means of tools from real and complex analysis, in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.

addition

increased by

{1, 2, 3, 4}

integers greater than zero and less than 5 form a set, as follows:

complex number

is a number that can be expressed in the form where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1. For example, −3.5 + 2i is a complex number. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i.

algebraic number

is any complex number that is a solution to some polynomial equation with rational coefficients; for example, every solution x of (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields, or shortly number fields.

The real number a of the complex number z = a + bi

is called the real part of z, and the real number b is often called the imaginary part. By this convention the imaginary part is a real number - not including the imaginary unit: hence b, not bi, is the imaginary part. (Others, however call bi the imaginary part.) The real part is denoted by Re(z) or ℜ(z), and the imaginary part b is denoted by Im(z) or ℑ(z). For example,

K+6, K+5, K+4 K+3.........answer is K+3

the smallest of four sonsecutive whole numbers, the biggest of which is K+6

subtraction

less than

addition

more than

polynomial

more than one term (5x+4 contains two)

T+9

number T increased by 9

(x-12)/40

number X decreased by 12 divided by forty

Numerals

number symbols

In Diophantine geometry

one asks whether there are any rational points (points all of whose coordinates are rationals) or integral points (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed.

monomial

one term (5x or 4)

addition

plus

multiplication

product

16(5+R)

product of 16 and the sum of 5 and number R

division

quotient

subtraction

remainder

Q-16

sixteen less than number Q

Algebraic number theory

studies algebraic properties and algebraic objects of interest in number theory. (Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.) A key topic is that of the algebraic numbers, which are generalizations of the rational numbers.

difference

subtraction

addition

sum

F, F+1, F+2.......answer is F+2

the greatest of 3 consecutive whole numbers, the smallest of which is F

constant

the number without a variable (5m+2). In this case, 2

magnitude

the relative greatness of positive and negative numbers

addition

total