Algebra 1, 2 Help, Worksheets, Expressions, Pre, College, Problems: What Is Algebra?
Algebra and why it is important, Algebra Expressions, problems and history
What is Algebra meaning and use
Algebra is a branch of mathematics that deals with properties of operations and the structures these operations are defined on. Elementary Algebra that follows the study of arithmetic is mostly occupied with operations on sets of whole and rational numbers and solving first and second order equations. What puts elementary algebra a step ahead of elementary arithmetic is a systematic use of letters to denote generic numbers.
Mastering of elementary algebra which is often hailed as a necessary preparatory step for the study of Calculus, is as often an insurmountable block in many a career. However, the symbolism that is first introduced in elementary algebra permeates all of mathematics. This symbolism is the alphabet of the mathematical language.
The word "algebra" is a shortened misspelled transliteration of an Arabic title al-jebr w'al-muqabalah (circa 825) by the Persian mathematician known as al-Khowarismi. The al-jebr part means "reunion of broken parts", the second part al-muqabalah translates as "to place in front of, to balance, to oppose, to set equal." Together they describe symbol manipulations common in algebra: combining like terms, moving a term to the other side of an equation, etc.
In its English usage in the 14th century, algeber meant "bone-setting," close to its original meaning. By the 16th century, the form algebra appeared in its mathematical meaning. Robert Recorde (c. 1510-1558), the inventor of the symbol "=" of equality, was the first to use the term in this sense. He, however, still spelled it as algeber. The misspellers proved to be more numerous, and the current spelling algebra took roots.
Thus the original meaning of algebra refers to what we today call elementary algebra which is mostly occupied with solving simple equations. More generally, the term algebra encompasses nowadays many other fields of mathematics: geometric algebra, abstract algebra, boolean algebra, to name a few.
Simply put, Algebra is about finding the unknown or it is about putting real life problems into equations and then solving them. Unfortunately many textbooks go straight to the rules, procedures and formulas, forgetting that these are real life problems being solved.
What is Algebra? Why Take Algebra?
A branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc.
We hear a lot about the importance that all children master algebra before they graduate from high school. But what exactly is algebra, and is it really as important as everyone claims? And why do so many people find it hard to learn?
Answering these questions turns out to be a lot easier than, well, answering a typical school algebra question, yet surprisingly, few people can give good answers.
First of all, algebra is not “arithmetic with letters.” At the most fundamental level, arithmetic and algebra are two different forms of thinking about numerical issues.
Let’s start with arithmetic. This is essentially the use of the four numerical operations addition, subtraction, multiplication, and division to calculate numerical values of various things. It is the oldest part of mathematics, having its origins in Sumeria (primarily today’s Iraq) around 5,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction of money as a means to measure an individual’s wealth and mediate the exchange of goods and services. The monetary tokens eventually gave way to abstract markings on clay tablets, which we recognize today as the first numerals (symbols for numbers). Over time, those symbols acquired an abstract meaning of their own: numbers. In other words, numbers first arose as money, and arithmetic as a means to use money in trade.
It should be noticed that counting predates numbers and arithmetic. Humans started to count things (most likely family members, animals, seasons, possessions, etc.) at least 6,000 years ago, as evidenced by the discovery of bones with tally marks on them, which anthropologists conclude were notched to provide what we would today call a numerical record. But those early humans did not have numbers, nor is there any evidence of any kind of arithmetic. The tally markers themselves were the record; the marks referred directly to things in the world, not to abstract numbers.
Something else to note is that arithmetic does not have to be done by the manipulation of symbols, the way we are taught today. The modern approach was developed over many centuries, starting in India in the early half of the First Millennium, adopted by the Arabic speaking traders in the second half of the Millennium, and then transported to Europe in the 13th Century. (Hence its present-day name “Hindu-Arabic arithmetic.”) Prior to the adoption of symbol-based, Hindu-Arabic arithmetic, traders performed their calculations using a sophisticated system of finger counting or a counting board (a board with lines ruled on it on which small pebbles were moved around). Arithmetic instruction books described how to calculate using words, right up to the 15th Century, when symbol manipulation began to take over.
Many people find arithmetic hard to learn, but most of us succeed, or at least pass the tests, provided we put in enough practice. What makes it possible to learn arithmetic is that the basic building blocks of the subject, numbers, arise naturally in the world around us, when we count things, measure things, buy things, make things, use the telephone, go to the bank, check the scores, etc. Numbers may be abstract — you never saw, felt, heard, or smelled the number 3 — but they are tied closely to all the concrete things in the world we live in.
With algebra, however, you are one more step removed from the everyday world. Those x’s and y’s that you have to learn to deal with in algebra denote numbers, but usually numbers in general, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction. Doing so requires quite a lot of effort and training.
The important thing to realize is that doing algebra is a way of thinking and that it is a way of thinking that is different from arithmetical thinking. Those formulas and equations, involving all those x’s and y’s, are merely a way to represent that thinking on paper. They no more are algebra than a page of musical notation is music. It is possible to do algebra without symbols, just as you can play and instrument without being ably to read music. In fact, traders and other people who needed it used algebra for 3,000 years before the symbolic form was introduced in the 16th Century. (That earlier way of doing algebra is nowadays referred to as “rhetorical algebra,” to distinguish it from the symbolic approach common today.)
There are several ways to come to an understanding of the difference between arithmetic and (school) algebra.
First, algebra involves thinking logically rather than numerically.
In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.
Arithmetic involves quantitative reasoning with numbers; algebra involves qualitative reasoning about numbers.
In arithmetic, you calculate a number by working with the numbers you are given; in algebra, you introduce a term for an unknown number and reason logically to determine its value.
The above distinctions should make it clear that algebra is not doing arithmetic with one or more letters denoting numbers, known or unknown.
For example, putting numerical values for a, b, c in the familiar formula
in order to find the numerical solutions to the quadratic equation
is not algebra, it is arithmetic.
In contrast, deriving that formula in the first place is algebra. So too is solving a quadratic equation not by the formula but by the standard method of “completing the square” and factoring.
When students start to learn algebra, they inevitably try to solve problems by arithmetical thinking. That’s a natural thing to do, given all the effort they have put into mastering arithmetic, and at first, when the algebra problems they meet are particularly simple (that’s the teacher’s classification as “simple”), this approach works.
In fact, the stronger a student is at arithmetic, the further they can progress in algebra using arithmetical thinking. For example, many students can solve the quadratic equation x2 = 2x + 15 using basic arithmetic, using no algebra at all.
Paradoxically, or so it may seem, however, those better students may find it harder to learn algebra. Because to do algebra, for all but the most basic examples, you have to stop thinking arithmetically and learn to think algebraically.
Is mastery of algebra (i.e., algebraic thinking) worth the effort? You bet — though you’d be hard pressed to reach that conclusion based on what you will find in most school algebra textbooks. In today’s world, algebra, and not arithmetic, should be the main goal of school mathematics instruction. For example, you need to use algebraic thinking if you want to write a macro to calculate the cells in a spreadsheet like Microsoft Excel. With a spreadsheet, you don’t need to do the arithmetic; the computer does it, generally much faster and with greater accuracy than any human can. What you, the person, have to do is create that spreadsheet in the first place. The computer can’t do that for you.
It doesn’t matter whether the spreadsheet is for calculating scores, keeping track of your finances, or running a business, you need to think algebraically to set it up to do what you want. That means thinking about or across numbers in general, rather than in terms of (specific) numbers.
An algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign.
3x2 + 2y + 7xy + 5
In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has four terms, 3x2, 2y, 7xy, and 5. Terms may consist of variables and coefficients, or constants.
In algebraic expressions, letters represent variables. These letters are actually numbers in disguise. In this expression, the variables are x and y. We call these letters "variables" because the numbers they represent can vary—that is, we can substitute one or more numbers for the letters in the expression.
Coefficients are the number part of the terms with variables. In 3x2 + 2y + 7xy + 5, the coefficient of the first term is 3. The coefficient of the second term is 2, and the coefficient of the third term is 7.
If a term consists of only variables, its coefficient is 1.
Constants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes, since there are no variables in the term that can change its value. In the expression 7x2 + 3xy + 8 the constant term is "8."
In algebra, we work with the set of real numbers, which we can model using a number line.
Real numbers describe real-world quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal. They can also be either rational or irrational. Numbers that are not "real" are called imaginary. Imaginary numbers are used by mathematicians to describe numbers that cannot be found on the number line. They are a more complex subject than we will work with here.
We call the set of real integers and fractions "rational numbers." Rational comes from the word "ratio" because a rational number can always be written as the ratio, or quotient, of two integers.
Examples of rational numbers
The fraction ½ is the ratio of 1 to 2.
Since three can be expressed as three over one, or the ratio of 3 to one, it is also a rational number.
The number "0.57" is also a rational number, as it can be written as a fraction.
Some real numbers can't be expressed as a quotient of two integers. We call these numbers "irrational numbers". The decimal form of an irrational number is a non-repeating and non-terminating decimal number. For example, you are probably familiar with the number called "pi". This irrational number is so important that we give it a name and a special symbol!
Pi cannot be written as a quotient of two integers, and its decimal form goes on forever and never repeats.
Translating Words into Algebra Language
Here are some statements in English. Just below each statement is its translation in algebra.
the sum of three times a number and eight
3x + 8
The words "the sum of" tell us we need a plus sign because we're going to add three times a number to eight. The words "three times" tell us the first term is a number multiplied by three.
In this expression, we don't need a multiplication sign or parenthesis. Phrases like "a number" or "the number" tell us our expression has an unknown quantity, called a variable. In algebra, we use letters to represent variables.
the product of a number and the same number less 3
x(x – 3)
The words "the product of" tell us we're going to multiply a number times the number less 3. In this case, we'll use parentheses to represent the multiplication. The words "less 3" tell us to subtract three from the unknown number.
a number divided by the same number less five
The words "divided by" tell us we're going to divide a number by the difference of the number and 5. In this case, we'll use a fraction to represent the division. The words "less 5" tell us we need a minus sign because we're going to subtract five.
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers (constants), variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic symbols. The use of expressions can range from the simple:
to the complex:
We can think of algebraic expressions as generalizations of common arithmetic operations that are formed by combining numbers, variables, and mathematical operations. Some common examples follow:
Linear expression: .
Quadratic expression: .
Rational expression: .
Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example:
would not be considered a mathematical expression but only a meaningless jumble.
In algebra an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression; the determination of this value depends on the semantics attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator to designate an internal direct sum.
Being an expression is a syntactic concept; although different mathematical fields have different notions of valid expressions, the values associated to variables does not play a role. See formal language for general considerations on how expressions are constructed, and formal semantics for questions concerning attaching meaning (values) to expressions.
Many mathematical expressions include letters called variables. Any variable can be classified as being either a free variable or a bound variable.
For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a function whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.
For example, the expression
evaluated for x = 10, y = 5, will give 2; but is undefined for y = 0.
The evaluation of an expression is dependent on the definition of the mathematical operators and on the system of values that is its context.
Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function. Example:
has free variable x, bound variable n, constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent with the simpler expression 12x. The value for x = 3 is 36.
The '+' and '−' (addition and subtraction) symbols have their usual meanings. Division can be expressed either with the '/' or with a horizontal dash. Thus
are perfectly valid. Also, for multiplication one can use the symbols '×' or a '•' (mid dot), or else simply omit it (multiplication is implicit); so:
are all acceptable. However, notice in the first example above how the "times" symbol resembles the letter 'x' and also how the '•' symbol resembles a decimal point, so to avoid confusion it's best to use one of the later two forms.
An expression must be well-formed. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic.
Expressions and their evaluation were formalised by Alonzo Church and Stephen Kleene in the 1930s in their lambda calculus. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages.
One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable. This is also true of any expression in any system that has power equivalent to the lambda calculus.
History of Algebra
The roots of algebra can be traced to the ancient Babylonians, who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, though this would not be realized until the medieval Muslim mathematicians.
By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them. Diophantus (3rd century AD), sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations.
The word algebra comes from the Arabic language (الجبر al-jabr "restoration") and much of its methods from Arabic/Islamic mathematics. Earlier traditions discussed above had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.
The Hellenistic mathematicians Hero of Alexandria and Diophantus as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level. For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he has to distinguish several types of equations.
In 1545, the Italian mathematician Girolamo Cardano published Ars magna -The great art, a 40-chapter masterpiece in which he gave for the first time a method for solving the general quartic equation.
The Greek mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead. Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to, and that he gave an exhaustive explanation of solving quadratic equations, supported by geometric proofs, while treating algebra as an independent discipline in its own right. His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".
The Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations. He also developed the concept of a function. The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji, and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.
François Viète’s work at the close of the 16th century marks the start of the classical discipline of algebra. In 1637, René Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph Lagrange in his 1770 paper Réflexions sur la résolution algébrique des équations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.
Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues. The "modern algebra" has deep nineteenth-century roots in the work, for example, of Richard Dedekind and Leopold Kronecker and profound interconnections with other branches of mathematics such as algebraic number theory and algebraic geometry. George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).
10 reasons why algebra is so important
"Numbers are everywhere, but the value of algebra is not that obvious. If you do something once, you will not need algebra. Algebra works best when you need to generalize your findings from a situation" (Usiskin, 1995).
The reasons why Algebra is important:
•One needs algebra as a requirement for the higher education acceptance.
•Without Algebra, you do not have the control of your own life but have to rely on others and that increases your chance of being fooled or decreases your chance of having a better life standard. i.e., in finance by earning more money, in stock market by analyzing the market.
•You may not learn much from more advanced disciplines such as Chemistry, Physics, etc.
•Algebra helps you share your knowledge with other people in an efficient way, and that includes teaching it.
•You save time with Algebra so that you do not waste time solving the same problem over and over again in different occasions.
•Algebra helps you become an active producer rather than a passive consumer. i.e. you can be the one who designs the formulae for others and make profit from it or the one who pays for it. i.e. there are people who sell the spreadsheet templates they produced.
•You enjoy life by adding meaningful experiences to your life. i.e. you can watch nature or everyday happenings and make recreational, meaningful calculations.
• Maths helps you become more sensitive to details, thus become a better problem solver. Life is full of problems, and analytical thinking is best achieved by studying maths.
•Maths works as an entertainment and recreation, and an exercise for your brain.
•Algebra helps to generalize your solutions to math related problems.
10 Everyday Reasons Why Algebra is Important in your Life
Mathematics is one of the first things you learn in life. Even as a baby you learn to count. Starting from that tiny age you will start to learn how to use building blocks how to count and then move on to drawing objects and figures. All of these things are important preparation to doing algebra.
The key to opportunity
These are the years of small beginnings until the day comes that you have to be able to do something as intricate as algebra. Algebra is the key that will unlock the door before you. Having the ability to do algebra will help you excel into the field that you want to specialize in. We live in a world where only the best, the daring and the humble succeed.
Taking a detour on not
Having the ability and knowledge to do algebra will determine whether you will take the short cut or the detour in the road of life. In other words, ample opportunities or career choices to decide from or limited positions with a low annual income.
Prerequisite for advanced training
Most employers expect their employees to be able to do the fundamentals of algebra. If you want to do any advanced training you will have to be able to be fluent in the concept of letters and symbols used to represent quantities.
When doing any form of science, whether just a project or a lifetime career choice, you will have to be able to do and understand how to use and apply algebra.
Every day life
Formulas are a part of our lives. Whether we drive a car and need to calculate the distance, or need to work out the volume in a milk container, algebraic formulas are used everyday without you even realizing it.
When it comes to analyzing anything, whether the cost, price or profit of a business you will need to be able to do algebra. Margins need to be set and calculations need to be made to do strategic planning and analyzing is the way to do it.
What about the entering of any data. Your use of algebraic expressions and the use of equations will be like a corner stone when working with data entry. When working on the computer with spreadsheets you will need algebraic skills to enter, design and plan.
Decisions like which cell phone provider gives the best contracts to deciding what type of vehicle to buy, you will use algebra to decide which one is the best one. By drawing up a graph and weighing the best option you will get the best value for your money.
How much can you earn on an annual basis with the correct interest rate. How will you know which company gives the best if you can't work out the graphs and understand the percentages. In today's life a good investment is imperative.
Writing of assignments
When writing any assignments the use of graphs, data and math will validate your statements and make it appear more professional. Professionalism is of the essence if you want to move ahead and be taken seriously.
Can you see the importance of algebra? Your day can be made a lot easier with planning. In financial decisions this can save you a lot of finances or maybe get you the best price available. It all comes down to planning and using the knowledge and algebraic skills you have to benefit your own life.
Use the key you have and make your life a lot smoother.