R3 – Logs
Definition This post covers the definition of R3 logs, logarithm formulae and provides some examples. Wikipedia defines logs as follows: ‘In mathematics, logarithm (log) is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.’ Logs are useful in expressing […]
R3 – Exponentiation – Fractions, Flip Sign and Rules
This post covers the rules of R3 exponentiation. It also describes the use of fractions and flip sign with exponentiation along with some examples. Fractions Exponents in fractions represent multiplication roots. The post on R3 roots outlined how the Counter and Full roots of Opposite Values are always equal and how there is only 1 […]
R3 – Exponentiation – Definition
Introduction This post covers the definition of R3 Exponentiation in Wave Numbers. In Wikipedia, exponentiation is defined as follows: ‘Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as “b raised to the power of n”. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases: bn = […]
R3 – Cube Roots
Similar to R3 Division, cube roots in R3 are solved using simultaneous equations. Simultaneous Equations Needed The following is a method for finding R3 cube roots using simultaneous equations. For example, take the cubic equation: As described in an earlier post, the square of the expression (a? + bi? + cj?)2 is (a2? + b2i2 + c2j2). This allows […]
R3 – Square Roots
Definition The square root of an expression in R3 is made of the roots of the individual Opposite Values with the signs remaining constant. Square of an Expression Given the generic expression (a? + bi? + cj?): Note that as a result of the Zero Rules of Rotation a??b?? + b??a?? = 0, so This […]
R3 – Roots – Definition
This post covers R3 roots including their derivation. Definition Wikipedia defines the square root as ‘a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x.’ Wave Numbers uses this definition and adapts it for roots of higher degrees. The root of an Opposite Value […]
R3 – Division – Examples
Next: Previous:
R3 – Division – Simultaneous Equations
Why Simultaneous Equations are Needed According to the theorem of multiplicative inverses, R3 lacks multiplicative inverses. Therefore, the Wave Number system employs an alternative method to compute the result of division in R3, using simultaneous equations. Consider the following equation: To obtain the result of the division, you need to solve the following equation: The […]
R3 – Reciprocals and Multiplicative Inverses
This post covers multiplicative inverses and the reciprocals of Opposite Values in R3. Need for Multiplicative Inverses As stated in the theorem in an earlier post, multiplicative inverses are not available in R3. Multiplicative inverses do not exist, as multiplication is not commutative in R3. The main use of the multiplicative inverse is to avoid […]
R3 – Division – Self Division and Unitaries
This post covers self division in R3. It looks at the cases where the result of division is a unitary and where it is not. Self Division In quaternions, R1 and R2, the division of an expression by itself results in the answer 1^. For example in R2: However, in Wave Numbers R3, division of […]