ALGEBRA REFERENCE
- Basic identities
- Arithmetic properties
- The associative property
- The commutative property
- The distributive property
- Properties of exponents
- Radicals
- Definition of a radical
- Properties of radicals
- Important constants
- Euler's number
- Pi
- Logarithms
- Definition of a logarithm
- Properties of logarithms
- Factoring equivalencies
- The quadratic formula
- Sequences
- Arithmetic sequences
- Geometric sequences
- Factorials
- Definition of a factorial
- Strange factorials
- Solving simultaneous equations
- Substitution method
- Addition method
- Contributors
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Basic identities
Note: while division by zero is popularly thought to be equal to infinity, this is not technically true. In some practical applications it may be helpful to think the result of such a fraction approaching positive infinity as a positive denominator approaches zero (imagine calculating current I=E/R in a circuit with resistance approaching zero -- current would approach infinity), but the actual fraction of anything divided by zero is undefined in the scope of either real or complex numbers.
Arithmetic properties
The associative property
In addition and multiplication, terms may be arbitrarily associated with each other through the use of parentheses:
The commutative property
In addition and multiplication, terms may be arbitrarily interchanged, or commutated:
The distributive property
Properties of exponents
Radicals
Definition of a radical
When people talk of a "square root," they're referring to a radical with a root of 2. This is mathematically equivalent to a number raised to the power of 1/2. This equivalence is useful to know when using a calculator to determine a strange root. Suppose for example you needed to find the fourth root of a number, but your calculator lacks a "4th root" button or function. If it has a yx function (which any scientific calculator should have), you can find the fourth root by raising that number to the 1/4 power, or x0.25.
It is important to remember that when solving for an even root (square root, fourth root, etc.) of any number, there are two valid answers. For example, most people know that the square root of nine is three, but negative three is also a valid answer, since (-3)2 = 9 just as 32 = 9.
Properties of radicals
Important constants
Euler's number
Euler's constant is an important value for exponential functions, especially scientific applications involving decay (such as the decay of a radioactive substance). It is especially important in calculus due to its uniquely self-similar properties of integration and differentiation.e approximately equals: 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996
Pi
Pi (π) is defined as the ratio of a circle's circumference to its diameter.Pi approximately equals: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511
Note: For both Euler's constant (e) and pi (π), the spaces shown between each set of five digits have no mathematical significance. They are placed there just to make it easier for your eyes to "piece" the number into five-digit groups when manually copying.
Logarithms
Definition of a logarithm
"log" denotes a common logarithm (base = 10), while "ln" denotes a natural logarithm (base = e).
Properties of logarithms
These properties of logarithms come in handy for performing complex multiplication and division operations. They are an example of something called a transform function, whereby one type of mathematical operation is transformed into another type of mathematical operation that is simpler to solve. Using a table of logarithm figures, one can multiply or divide numbers by adding or subtracting their logarithms, respectively. then looking up that logarithm figure in the table and seeing what the final product or quotient is.
Slide rules work on this principle of logarithms by performing multiplication and division through addition and subtraction of distances on the slide.
Marks on a slide rule's scales are spaced in a logarithmic fashion, so that a linear positioning of the scale or cursor results in a nonlinear indication as read on the scale(s). Adding or subtracting lengths on these logarithmic scales results in an indication equivalent to the product or quotient, respectively, of those lengths.
Most slide rules were also equipped with special scales for trigonometric functions, powers, roots, and other useful arithmetic functions.
Factoring equivalencies
The quadratic formula
Sequences
Arithmetic sequences
An arithmetic sequence is a series of numbers obtained by adding (or subtracting) the same value with each step. A child's counting sequence (1, 2, 3, 4, . . .) is a simple arithmetic sequence, where the common difference is 1: that is, each adjacent number in the sequence differs by a value of one. An arithmetic sequence counting only even numbers (2, 4, 6, 8, . . .) or only odd numbers (1, 3, 5, 7, 9, . . .) would have a common difference of 2.In the standard notation of sequences, a lower-case letter "a" represents an element (a single number) in the sequence. The term "an" refers to the element at the nth step in the sequence. For example, "a3" in an even-counting (common difference = 2) arithmetic sequence starting at 2 would be the number 6, "a" representing 4 and "a1" representing the starting point of the sequence (given in this example as 2).
A capital letter "A" represents the sum of an arithmetic sequence. For instance, in the same even-counting sequence starting at 2, A4 is equal to the sum of all elements from a1 through a4, which of course would be 2 + 4 + 6 + 8, or 20.
Geometric sequences
A geometric sequence, on the other hand, is a series of numbers obtained by multiplying (or dividing) by the same value with each step. A binary place-weight sequence (1, 2, 4, 8, 16, 32, 64, . . .) is a simple geometric sequence, where the common ratio is 2: that is, each adjacent number in the sequence differs by a factor of two.
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