Project in Mathematics

Mathematics mini-booklet

Submitted to:Mrs. Jennife Rearte

GROUP MEMBERS

MACKYLAH YAUDER JAN PAULENE SAMORANOS VAN DEVEN GALOS DANIEL CARL GARL GARRIDO LEADER ASSISTANT LEDER

MARC LOUISE DELLOSA GERIC PALENCIA MHIKO ZUBIA ILLEANA DELOSARIO

DANIELA ANN GOMEZ ANGELIKA GREFIL AHSLEY NICOLE LEOCADIO ERICKA GRACE MANIPON

TABLE OF CONTENTSUnit 1:

Mathematical terms................... 3

Unit 2:

Great Mathematicians.................6

Unit 3:

Lesson 1-Whole Numbers.............8

Lesson 2-Whole Number Operations.........................................9

Lesson 3-Decimal....................10

Lesson 4-Fraction....................11

Lesson 5-Geometry.....................11

Lesson 6-Integers......................13

Lesson 7-Algebra.........................14

Unit 1Mathematical termsAcute angle – An angle which measures below 90°.

Acute triangle – A triangle containing only acute angles.

Additive inverse – The opposite of a number or its negative. A number plus its additive inverse equals 0.

Adjacent angles – Angles with a common side and vertex.

Angle – Created by two rays and containing an endpoint in common.

Arc – A set of points that lie on a circle and that are positioned within a central angle.

Area – The space contained within a shape.

Average – The numerical result of dividing the sum of two or more quantities by the number of quantities.

Binomial – An expression in algebra that consists of two terms.

Bisect – To divide into two equal sections.

Canceling – In multiplication of fractions, when one number is divided into both a numerator and a denominator.

Cartesian coordinates – Ordered number pairs that are assigned to points on a plane.

Chord – A line segment that connects two points on a circle.

Circle – A set of points that are all the same distance from a given point.

Circumference – The distance measured around a circle.

Coefficient – A number that is placed in front of a variable. For example, in 6x, 6 is the coefficient.

Common denominator – A number that can be divided evenly by all denominators in the problem.

Complementary angles – Two angles in which the sum of their measurements equals 90°.

Complex fraction – A fraction that contains a fraction or fractions in the numerator and/or denominator.

Congruent – Exactly the same. Identical in regard to size and shape.

Coordinate graph – Two perpendicular number lines, the x axis and the y axis, which make a plane upon which each point is assigned a pair of numbers.

Cube – A solid with six sides, with the sides being equal squares and the edges being equal. Also, the resulting number when a number is multiplied by itself twice.

Cube root – A number that when multiplied by itself twice gives the original number. For example, 4 is the cube root of 64.

Decimal fraction – Fraction with a denominator of 10, 100, 1,000, etc., written using a decimal point.

Degree – The measurement unit of an angle.

Denominator – The bottom symbol or number of a fraction.

Diameter – A line segment that contains the center and has its endpoints on the circle. Also, the length of this segment.

Difference – That which results from subtraction.

Equation – A relationship between symbols and/or numbers that is balanced.

Equilateral triangle – A triangle that has three equal angles and three sides the same length.

Even number – An integer which can be divided by 2, with no remainder.

Expanded notation – To point out the place value of a digit by writing the number as the digit times its place value.

Exponent – A positive or negative number that expresses the power to which the quantity is to be raised or lowered. It is placed above and to the right of the number.

Exterior angle – In a triangle, an exterior angle is equal to the measures of the two interior angles added together.

Factor – As a noun, it is a number or symbol which divides evenly into a larger number. As a verb, it means to find two or more values whose product equals the original value.

F.O.I.L. Method – A method used for multiplying binomials in which the first terms, the outside terms, the inside terms, and then the last terms are multiplied.

Fraction – A symbol which expresses part of a whole. It contains a numerator and a denominator.

Greatest common factor – The largest factor that is common to two or more numbers.

Hypotenuse – In a right triangle it is the side opposite from the 90° angle.

Imaginary number – The square root of a negative number.

Improper fraction – A fraction in which the numerator is larger than the denominator.

Integer – A whole number. It may be positive, negative, or zero.

Interior angles – Angles formed inside the shape or inside two parallel lines.

Intersecting lines – Lines that come together at a point.

Interval – The numbers that are contained within two specific boundaries.

Irrational number – A number that is not rational (cannot be written as a fraction x/y, with x a natural number and y an integer).

Isosceles triangle – A triangle with two equal sides and two equal angles across from them.

Least common multiple – The smallest multiple that is common to two or more numbers.

Linear equation – An equation where the solution set forms a straight line when it is plotted on a coordinate graph.

Lowest common denominator – The smallest number that can be divided evenly by all denominators in the problem.

Mean – The average of a number of items in a group (total the items and divide by the number of items).

Median – The middle item in an ordered group. If the group has an even number of items, the median is the average of the two middle terms.

Mixed number – A number containing both a whole number and a fraction.

Monomial – An expression in algebra that consists of only one term.

Natural number – A counting number.

Negative number – A number less than zero.

Nonlinear equation – An equation where the solution set does not form a straight line when it is plotted on a coordinate graph.

Number line – A visual representation of the positive and negative numbers and zero.

Numerator – The top symbol or number of a fraction.

Obtuse angle – An angle which is larger than 90° but less than 180°.

Obtuse triangle – A triangle which contains an obtuse angle.

Odd number – An integer (whole number) that is not divisible evenly by 2.

Ordered pair – Any pair of elements (x,y) where the first element is x and the second element is y. These are used to identify or plot points on coordinate graphs.

Origin – The intersection point of the two number lines of a coordinate graph. The intersection point is represented by the coordinates (0,0).

Parallel lines – Two or more lines which are always the same distance apart. They never meet.

Percentage – A common fraction with 100 as its denominator.

Perpendicular lines – Two lines which intersect at right angles.

Pi (π) – A constant that is used for determining the circumference or area of a circle. It is equal to approximately 3.14.

Polynomial – An expression in algebra that consists of two or more terms.

Positive number – A number greater than zero.

Power – A product of equal factors. 3 x 3 x 3 = 33, read as “three to the third power” or “the third power of three.” Power and exponent can be used interchangeably.

Prime number – A number that can be divided by only itself and one.

Proper fraction – A fraction in which the numerator is less than the denominator.

Proportion – Written as two equal ratios. For example, 5 is to 4 as 10 is to 8, or 5/4 = 10/8.

Pythagorean theorem – A theorem concerning right triangles. It states that the sum of the squares of a right triangle’s two legs is equal to the square of the hypotenuse (a2 + b2 = c2).

Quadrants – The four divisions on a coordinate graph.

Quadratic equation – An equation that may be expressed as Ax2 + Bx + C = 0.

Radical sign – A symbol that designates a square root.

Radius – A line segment where the endpoints lie one at the center of a circle and one on the circle. The term also refers to the length of this segment.

Ratio – A comparison between two numbers or symbols. May be written x:y, x/y, or x is to y.

Rational number – An integer or fraction such as 7/7 or 9/4 or 5/1. Any number that can be written as a fraction x/y with x a natural number and y an integer.

Reciprocal – The multiplicative inverse of a number. For example, 2/3 is the reciprocal of 3/2.

Reducing – Changing a fraction into its lowest terms. For example, 3/6 is reduced to ½.

Right angle – An angle which measures 90°.

Right triangle – A triangle which contains a 90° angle.

Scalene triangle – A triangle in which none of the sides or angles are equal.

Scientific notation – A number between 1 and 10 and multiplied by a power of 10. Used for writing very large or very small numbers.

Set – A group of objects, numbers, etc.

Simplify – To combine terms into fewer terms.

Solution, or Solution set – The entirety of answers that may satisfy the equation.

Square – The resulting number when a number is multiplied by itself. Also, a four-sided figure with equal sides and four right angles. The opposite sides are parallel.

Square root – The number which when multiplied by itself gives you the original number. For example, 6 is the square root of 36.

Straight angle – An angle which is equal to 180°.

Straight line – The shortest distance between two points. It continues indefinitely in both directions.

Supplementary angles – Two angles that when combined the sum equals 180°.

Term – A literal or numerical expression that has its own sign.

Transversal – A line which crosses two or more parallel or nonparallel lines in a plane.

Triangle – A three-sided closed figure. It contains three angles that when combined the sum equals 180°.

Trinomial – An expression in algebra which consists of three terms.

Unknown – A symbol or letter whose value is unknown.

Variable – A symbol that stands for a number.

Vertical angles – The opposite angles that are formed by the intersection of two lines. Vertical angles are equal.

Volume – The amount which can be held, as measured in cubic units. The volume of a rectangular prism = length times width times height.

Whole number – 0, 1, 2, 3, 4, 5, 6, 7, 8, etc.

Unit 2

Great Mathematicians

Pythagoras of Samos

Greek Mathematician Pythagoras is considered by some to be one of the first great mathematicians. Living around 570 to 495 BC, in modern day Greece, he is known to have founded the Pythagorean cult, who were noted by Aristotle to be one of the first groups to actively study and advance mathematics. He is also commonly credited with the Pythagorean Theorem within trigonometry. However, some sources doubt that is was him who constructed the proof (Some attribute it to his students, or Baudhayana, who lived some 300 years earlier in India). Nonetheless, the effect of such, as with large portions of fundamental mathematics, is commonly felt today, with the theorem playing a large part in modern measurements and technological equipment, as well as being the base of a large portion of other areas and theorems in mathematics. But, unlike most ancient theories, it played a bearing on the development of geometry, as well as opening the door to the study of mathematics as a worthwhile endeavor. Thus, he could be called the founding father of modern mathematics.

Euclid

Living around 300BC, he is considered the Father of Geometry and his magnum opus: Elements, is one the greatest mathematical works in history, with its being in use in education up until the 20th century. Unfortunately, very little is known about his life, and what exists was written long after his presumed death. Nonetheless, Euclid is credited with the instruction of the rigorous, logical proof for theorems and conjectures. Such a framework is still used to this day, and thus, arguably, he has had the greatest influence of all mathematicians on this list. Alongside his Elements were five other surviving works, thought to have been written by him, all generally on the topic of Geometry or Number theory. There are also another five works that have, sadly, been lost throughout history.

Leonhard Euler

If Gauss is the Prince, Euler is the King. Living from 1707 to 1783, he is regarded as the greatest mathematician to have ever walked this planet. It is said that all mathematical formulas are named after the next person after Euler to discover them. In his day he was ground breaking and on par with Einstein in genius. His primary (if that’s possible) contribution to the field is with the introduction of mathematical notation including the concept of a function (and how it is written as f(x)), shorthand trigonometric functions, the ‘e’ for the base of the natural logarithm (The Euler Constant), the Greek letter Sigma for summation and the letter ‘/i’ for imaginary units, as well as the symbol pi for the ratio of a circles circumference to its diameter. All of which play a huge bearing on modern mathematics, from the everyday to the incredibly complex. As well as this, he also solved the Seven Bridges of Koenigsberg problem in graph theory, found the Euler Characteristic for connecting the number of vertices, edges and faces of an object, and (dis)proved many well known theories, too many to list. Furthermore, he continued to develop calculus, topology, number theory, analysis and graph theory as well as much, much more – and ultimately he paved the way for modern mathematics and all its revelations. It is probably no coincidence that industry and technological developments rapidly increased around this time.

Rene Descartes

French Philosopher, Physicist and Mathematician Rene Descartes is best known for his ‘Cogito Ergo Sum’ philosophy. Despite this, the Frenchman, who lived 1596 to 1650, made ground breaking contributions to mathematics. Alongside Newton and Leibniz, Descartes helped provide the foundations of modern calculus (which Newton and Leibniz later built upon), which in itself had great bearing on the modern day field. Alongside this, and perhaps more familiar to the reader, is his development of Cartesian Geometry, known to most as the standard graph (Square grid lines, x and y axis, etc.) and its use of algebra to describe the various locations on such. Before this most geometers used plain paper (or another material or surface) to preform their art. Previously, such distances had to be measured literally, or scaled. With the introduction of Cartesian Geometry this changed dramatically, points could now be expressed as points on a graph, and as such, graphs could be drawn to any scale, also these points did not necessarily have to be numbers. The final contribution to the field was his introduction of superscripts within algebra to express powers. And thus, like many others in this list, contributed to the development of modern mathematical notation.

Unit 3

Lesson 1:

Whole NumbersIn mathematics, the natural numbers (sometimes called the whole numbers) are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".

Another use of natural numbers is for what linguists call nominal numbers, such as the model number of a product, where the "natural number" is used only for naming (as distinct from a serial number where the order properties of the natural numbers distinguish later uses from earlier uses) and generally lacks any meaning of number as used in mathematics but rather just shares the character set.

The natural numbers are a basis from which many other number sets may be built by extension: the integers, by including an unresolved negation operation; the rational numbers, by including with the integers an unresolved division operation; the real numbers by including with the rationals the termination of Cauchy sequences; the complex numbers, by including with the real numbers the unresolved square root of minus one; the hyperreal numbers, by including with real numbers the infinitesimal value epsilon; vectors, by including a vector structure with reals; matrices, by having vectors of vectors; the nonstandard integers; and so on. Thereby the natural numbers are canonically embedded (identification) in the other number systems.

Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.

There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1. Corresponding to the positive integers 1, 2, 3.

This distinction is of no fundamental concern for the natural numbers (even when viewed via additional axioms as semigroup with respect to addition and monoid for multiplication). Including the number 0 just supplies an identity element for the former (binary) operation to achieve a monoid structure for both, and a (trivial) zero divisor for the multiplication.

In common language, for example in grade school, natural numbers may be called counting numbers to distinguish them from the real numbers which are used for measurement.

Lesson 2: Whole Number Operations

Rounding Whole Numbers

Rounding is the process of finding the closest number to a specific value. You round a number up or down based on the last digit you are interested in.

For example, rounding the number 245 to the nearest tens place would round up to 250, while the number 324 rounded to the nearest tens place would be rounded down to 320.

Following the same logic, one could round to the nearest whole number. For example, 1.5 (pronounced as "one point five" or "one and a half") would be rounded up to 2, and 2.1 would be rounded down to 2.

Adding Whole Numbers

First, arrange the numbers in columns. For example, 134+937.

134

+937

Add the first column (starting on the right)

134

+937

1,071

Note the 10's digit put under the next column. Now add the next column and the number underneath:

Finish it off with the other columns:

So the answer to 134 + 937 is 1071

Subtracting Whole numbers

To subtract numbers think of a basket of oranges. If you have ten oranges in a basket and you remove eight oranges you are left with two oranges. For example:

If you have ten oranges in a basket and you remove all ten then you will no longer have any oranges so you are left with zero oranges. For example:

10

-10

0

To subtract large numbers use this method:

1. Arrange the number that is being subtracted from on top of the number being subtracted from it.(ex. 2594-1673)

2594

-1674

2. Subtract each column starting from the right and going to the left

2 5 9 4

-1 6 7 3

2 1

3. If you encounter a number that can't be subtracted without becoming negative,"borrow" subtract(if possible) 1 from the next digit over and add 10 to the digit that can't be subtracted(if not possible continue to borrow from the next digit).

1 15

X X 9 4

-1 6 7 3

________

9 2 1

4. continue until done

note: that 921+1673=2594.

Multiplying Whole Numbers

Single number times Single number producing a Single number.

Take the first number as 1. Take the second number as 2. Repeatedly add 1 for 2 times.

1 = 2

2 = 3

(1)x(2)= 2 x 3 = 2 + 2 + 2 = 6

Dividing Whole Numbers

Dividing whole numbers is the process of determining how many times one number, called the dividend, contains another number, called the divisor.

12 / 3

In this example, 12 is the dividend and 3 is the divisor. Performing a division gives a quotient.

8 / 4 = 2

In the above example, 4 goes into 8 twice; therefore, the quotient would be 2.

What happens when the dividend cannot be evenly split by the divisor? This leftover quantity is called the remainder. It's usually separated from the main part of the answer by a lowercase letter “r”.

13 / 5 = 2 r 3

Divisions are often represented as fractions. For example,

68 / 43 = 68 divided by 43

Some tips:

Any number that ends in 0, 2, 4, 6, or 8 can be divided by 2.

Any number that ends in 0 or 5 can be divided by 5.

Factoring Whole Numbers

Factoring is the process of determining what prime numbers (numbers that cannot be divided by any number but 1 and itself; 2,3, and 5 are prime numbers) when multiplied will give a specific number. This process of factoring is very important in reducing fractions, which is covered in the Fractions chapter of this book. For example:

4 = 2 x 2

Or a more complicated example:

180 = 2 x 2 x 3 x 3 x 5

Lesson 3: Decimal

The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.

Decimal notation often refers to a base-10 positional notation such as the Hindu-Arabic numeral system or rod calculus; however, it can also be used more generally to refer to non-positional systems such as Roman or Chinese numerals which are also based on powers of ten.

A decimal number, or just decimal, refers to any number written in decimal notation, although it is more commonly used to refer to numbers that have a fractional part separated from the integer part with a decimal separator (e.g. 11.25).

A decimal may be a terminating decimal, which has a finite fractional part (e.g. 15.600); a repeating decimal, which has an infinite (non-terminating) fractional part made up of a repeating sequence of digits (e.g. 5.8144); or an infinite decimal, which has

a fractional part that neither terminates nor has an infinitely repeating pattern (e.g. 3.14159265...). Decimal fractions have terminating decimal representations, whereas irrational numbers have infinite decimal representations.

Lesson 4: FractionA fraction (from Latin: fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: 17/3) consists of an integer numerator, displayed above a line (or before a slash), and a non-zero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates or 3/4 of a cake.

Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, or negative exponents (as in 0.01, 1%, and 10−2 respectively, all of which are equivalent to 1/100). An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1.

Other uses for fractions are to represent ratios and to represent division.Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and the division 3 ÷ 4 (three divided by four).

In mathematics the set of all numbers which can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. The test for a number being a rational number is that it can be written in that form (i.e., as a common fraction). However, the word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).

Lesson 5: GeometryGeometry (from the Ancient Greek: geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world,

both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

In Euclid's time, there was no clear distinction between physical and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, 'space' (whether 'point', 'line', or 'plane') lost its intuitive contents, so today one has to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meanings) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure which allow one to speak about length. Modern geometry has many ties to physics as is exemplified by the links between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

While the visual nature of geometry makes it initially more accessible than other mathematical areas such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).

Lesson 6: IntegersAn integer (from the Latin integer meaning "whole")[note 1] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5½, and √2 are not.

The set of integers consists of zero (0), the natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e. −1, −2, −3, ...). This is often denoted by a boldface Z ("Z") or blackboard bold (Unicode U+2124 ) standing for the German word ℤZahlen ([ˈtsaːlən], "numbers"). is a subset of the ℤsets of rational and real numbers and, like the natural numbers, is countably infinite.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the (rational) integers are the algebraic integers that are also rational numbers.

Lesson 7: AlgebraAlgebra (from Arabic al-jebr meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols and is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1048–1131).

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take

on many values. For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x=3. In E=mc^2, the letters E and m are variables, and the letter c is a constant. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called ”algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology.

A mathematician who does research in algebra is called an algebraist.

Different meanings of "Algebra"The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

As a single word without article, "algebra" names a broad part of mathematics.

As a single word with article or in plural, "algebra" denotes a specific mathematical structure. See algebra (ring theory) and algebra over a field. More generally, in universal algebra, it can refer to any structure.

With a qualifier, there is the same distinction:

Without article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).

With an article, it means an instance of some abstract structure, like a Lie algebra or an associative algebra.

Frequently both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.