Sequential Multiplication Operation

A Sequential Multiplication Operation is a sequential mathematical operation that involves arithmetic multiplications (where a multiplication operation is repeated a constant number of times).

• AKA: Π, Exponentiation.
• Context:
  • It can be represented as [math]\displaystyle{ x^a }[/math], where [math]\displaystyle{ x }[/math] is the Exponent Base and [math]\displaystyle{ a }[/math] is the Exponent.
  • It can range from being an Abstract Exponentiation Operation to being an Exponentiation Software Function.
  • It can (typically) have properties:
    • [math]\displaystyle{ x^a \times x^b = x^{(a + b)} }[/math].
    • [math]\displaystyle{ x^a \times y^a = (x \times y)^a }[/math].
    • [math]\displaystyle{ (x^a)^b = x^{(a \times b)} }[/math].
    • [math]\displaystyle{ x }[/math] ^(a/b) = b^th root of (x ^a) = (b^th (root)(x) ) ^a
    • [math]\displaystyle{ x }[/math] ^(-a) = 1 / [math]\displaystyle{ x }[/math] ^a
    • [math]\displaystyle{ x }[/math] ^{(a - b)} = x ^a / [math]\displaystyle{ x }[/math] ^b
  • It can be used in a Polynomial Function.
• Example(s):
  • a Squaring Operation.
  • [math]\displaystyle{ \cdot(1, 2, 3, 4) \equiv 24. }[/math]
  • [math]\displaystyle{ \prod_{i=1}^4 i \equiv 1\cdot 2\cdot 3\cdot 4 \equiv 24. }[/math]
• Counter-Example(s):
  • Sequential Summation.
• See: Partial Sum, Prefix Sum, Running Total, Polynomial, Abelian Group, Matrix Multiplication, Series Expansion, Nth Root Function, Scalar Numeric Function, Base (Exponentiation), Multiplication, Product (Mathematics), Superscript, Matrix Exponential, Linear Differential Equation, Compound Interest, Population Growth.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Multiplication#Products_of_sequences Retrieved:2015-1-17.
  • The product of a sequence of terms can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. Unicode position U+220F (∏) contains a glyph for denoting such a product, distinct from U+03A0 (Π), the letter. The meaning of this notation is given by: : [math]\displaystyle{ \prod_{i=1}^4 i = 1\cdot 2\cdot 3\cdot 4, }[/math] that is : [math]\displaystyle{ \prod_{i=1}^4 i = 24. }[/math] The subscript gives the symbol for a dummy variable (i in this case), called the "index of multiplication" together with its lower bound (1), whereas the superscript (here 4) gives its upper bound. The lower and upper bound are expressions denoting integers. The factors of the product are obtained by taking the expression following the product operator, with successive integer values substituted for the index of multiplication, starting from the lower bound and incremented by 1 up to and including the upper bound. So, for example: : [math]\displaystyle{ \prod_{i=1}^6 i = 1\cdot 2\cdot 3\cdot 4\cdot 5 \cdot 6 = 720 }[/math] More generally, the notation is defined as : [math]\displaystyle{ \prod_{i=m}^n x_i = x_m \cdot x_{m+1} \cdot x_{m+2} \cdot \,\,\cdots\,\, \cdot x_{n-1} \cdot x_n, }[/math] where m and n are integers or expressions that evaluate to integers. In case m = n, the value of the product is the same as that of the single factor x_m. If m > n, the product is the empty product, with the value 1.

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Exponentiation Retrieved:2014-10-16.
  • 'Exponentiation is a mathematical operation, written as bⁿ, involving two numbers, the base b and the exponent (or power) n. When n is a natural number (i.e., a positive integer), exponentiation corresponds to repeated multiplication of the base: that is, bⁿ is the product of multiplying n bases:  :[math]\displaystyle{ b^n = \underbrace{b \times \cdots \times b}_n }[/math]

    The exponent is usually shown as a superscript to the right of the base. The exponentiation bⁿ can be read as b raised to the n-th power, or b raised to the power of n, or b raised by the exponent of n, or most briefly as b to the n. The two most common exponents have their own words: the exponent 2 (or 2nd power) is called the square of b (b²) or b squared ; the exponent 3 (or 3rd power) is called the cube of b (b³) or b cubed.

    When n is a negative integer and b is not zero, bⁿ is naturally defined as 1/b⁻n, preserving the property .

    Exponentiation, where the exponent is a matrix, is used for solving systems of linear differential equations.

    Exponentiation is used extensively in many other fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.

2009

• http://www.math.com/tables/algebra/exponents.htm
  • [math]\displaystyle{ x }[/math] ^a [math]\displaystyle{ x }[/math] ^b = [math]\displaystyle{ x }[/math] ^{(a + b)}
  • [math]\displaystyle{ x }[/math] ^a [math]\displaystyle{ y }[/math] ^a = (xy) ^a
  • (x ^a) ^b = [math]\displaystyle{ x }[/math] ^(ab)
  • [math]\displaystyle{ x }[/math] ^(a/b) = b^th root of (x ^a) = (b^th (root)(x) ) ^a
  • [math]\displaystyle{ x }[/math] ^(-a) = 1 / [math]\displaystyle{ x }[/math] ^a
  • [math]\displaystyle{ x }[/math] ^{(a - b)} = x ^a / [math]\displaystyle{ x }[/math] ^b

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