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Order of operations, Adding/Subtracting negative numbers, Multiplying and dividing negative numbers, Adding and Subtracting fractions multiplying fractions, Dividing fractions, Level 1 Exponents, Level 2 Exponents, Level 3 Exponents, Negative Exponent Intuition, Exponent Rules Part, Simplifying radicals, Introduction to Logarithms, Unit Conversion, Speed translation, Introduction to logarithm properties, Scientific Notation |
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Algebra 2
Linear Equations, Solving Inequalities, graphing lines, Graphing linear equations Slope and Y-intercept intuition, the Slope of a line, Equation of a line, Averages, Integer sums, Taking percentages, growing by a percentage, systems of Equations, Ratios |
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Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of the main branches of pure mathematics. The part of algebra called elementary algebra is often part of the curriculum in secondary education and introduces the concept of variables representing numbers. Statements based on these variables are manipulated using the rules of operations that apply to numbers, such as addition. This can be done for a variety of reasons, including equation solving. Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations are devised for things other than numbers. Addition and multiplication can be generalised and their precise definitions lead to structures such as groups, rings and fields.
Elementary algebra
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as a, x, or y). This is useful because:
It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these (for instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a number x such that ax+b=c". Step which lets to the conclusion that is not the nature of the specific numbers the one that allows us to solve it but that of the operations involved).
It allows the formulation of functional relationships (such as "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied.").
Properties of operations
| Operation |
commutative |
associative |
identity element |
inverse operation |
Addition
(x+y) |
x + y = y + x |
(x + y) + z = x + (y + z) |
0,which preserves numbers: a + 0 = a |
Subtraction ( - ) |
Multiplication
x X y or x • y |
x X y = y X x |
(x X y) X z = xX (y X z) |
1, which preserves numbers: a × 1 = a |
Division
( / ) |
|
Exponentiation
ab |
Not commutative |
Not associative |
1, which preserves numbers: a1 = a |
Logarithm
(Log) |
- The operation of addition...
- has an inverse operation called subtraction: (a + b) − b = a, which is the same as adding a negative number, a − b = a + (−b);
- The operation of multiplication...
- means repeated addition: a × n = a + a +...+ a (n number of times);
- has an inverse operation called division that works for non-zero numbers: (ab)/b = a, which is the same as multiplying by a reciprocal, a/b = a(1/b);
- distributes over addition: (a + b)c = ac + bc;
- is abbreviated by juxtaposition: a × b ≡ ab;
- The operation of exponentiation...
- means repeated multiplication: an = a × a ×...× a (n number of times);
- has an inverse operation, called the logarithm: alogab = b = logaab;
- distributes over multiplication: (ab)c = acbc;
- can be written in terms of n-th roots: am/n ≡ (n√a)m and thus even roots of negative numbers do not exist in the real number system. (See: complex number system)
- has the property: abac = ab + c;
- has the property: (ab)c = abc.
- in general ab ≠ ba and (ab)c ≠ a(bc);
Order of operations
In mathematics it is important that the value of an expression is always computed the same way. Therefore, it is necessary to compute the parts of an expression in a particular order, known as the order of operations. The standard order of operations is expressed in the following chart.
-
-
- parenthesis
- exponents and roots
- multiplication and division
- addition and subtraction
A common mnemonic device for remembering this order is PEMDAS. Generally in Elementary Algebra, the use of brackets (often called parentheses) and their simple applications will be taught at most of the schools in the world.
Equations
Properties of equality
- The relation of equality (=) is...
- reflexive: a = a;
- symmetric: if a = b then b = a;
- transitive: if a = b and b = c then a = c.
- The relation of equality (=) has the property...
- that if a = b and c = d then a + c = b + d and ac = bd;
- that if a = b then a + c = b + c;
- that if two symbols are equal, then one can be substituted for the other.
Properties of inequality
- The relation of inequality (<) has the property...
- of transivity: if a < b and b < c then a < c;
- that if a < b and c < d then a + c < b + d;
- that if a < b and c > 0 then ac < bc;
- that if a < b and c < 0 then bc < ac.
In mathematics, a percentage is a way of expressing a number as a fraction of 100 ( per cent meaning "per hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct". For example, 65% (read as "sixty-five percent") is equal to 65 / 100, or 0.65.
Percentages are used to express how large/small one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity, which should be greater than zero.
Percentage is a very good technique to see how much a task has been completed. e.g a task takes 20 hours to be done completely. If the task is 70% done it means (80%=(x/20)*100 so x=80*20/100=16 hours) 16 hours work is completed and 4 hours work left.
Ratio
In mathematics, a ratio expresses the magnitude of quantities relative to each other. Specifically, the ratio of two quantities indicates how many times the first quantity is contained in the second and may be expressed algebraically as their quotient. Mathematically, a proportion is defined as the equality of two ratios. However in common usage the word proportion is used to indicate a ratio, especially the ratio of a part to a whole.
The ratio of quantities A and B can be expressed as:
- the ratio of A to B
- as B is to A
- A:B.
The quantities A and B are sometimes called terms with A being the antecedent and B being the consequent.
The proportion expressing the equality of the ratios A:B and C:D is written A:B=C:D or A:B::C:D. Again, A, B, C, D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion
Polynomials
A polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant non-negative integer exponent). For example, x2 + 2x − 3 is a polynomial in the single variable x.
An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.
Abstract algebra
Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts.
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property, specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups which are the group of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra. |
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