Mathematical English (a brief summary

The American Mathematical Monthly, 2007

Zenodo (CERN European Organization for Nuclear Research), 2022

Review of complex numbers. A complex number is any expression of the form x + iy where x and y are real numbers. x is called the real part and y is called the imaginary part of the complex number x + iy. The complex number x − iy is said to be complex conjugate of the number x + iy.

International Electronic Journal of Mathematics Education

This research investigated students' sources and causes of errors and misconception in solving routine problems involving addition, subtraction, multiplication, and division of integers. This qualitative study involved observation of eight Year 7 classes and interviews with the respective classroom teachers. Sixteen Year 7 students who exhibit errors when solving the problems given in the Error Identification Integer Test (EIIT) were also interviewed to probe their thinking. The different types of errors were categorized according to how they were construed. The sources of errors were found to stem from carelessness, poor basic knowledge such as the inability to multiply and divide even the whole numbers, inability to assimilate concepts of integers since they are used to the schema of whole numbers, and rule mix-up which is also the result of surface understanding. Teachers were questioned about common errors and possible reasons for these errors made by their students. The main cause of errors and misconceptions is superficial understanding, which was most probably due to teachers rushing to complete the extensive syllabus, and consequently, students resorted to memorizing rules because of surface understanding. Teaching episodes were found to lack multiple-representation, creativity, as well as cooperative learning and active learning.

Computing, 1979

Square Rooting Is as Difficult as Multiplication. It is shown that multiplication of numbers and square rooting have the same complexity, i.e. from a program for multiplication one can construct a program for square rooting with the same asymptotic time complexity (1 step ~ 1 bit-operation) and vice versa. It follows from the Schfnhage-Strassen algorithm that square rooting can be performed in 0 (n log n log log n) bit-operations. Die Komplexitiit des Wurzelziehens. Es wird gezeigt, dab Multiplikation von Zahlen und Bestimmen der Quadratwurzel yon gleicher Komplexit/it sind, d.h. aus einem Programm zur Multiplikation kann man eines zum Wurzelziehen konstruieren, das gr6Benordnungsm/~Big die gleiche Zeitkomplexit~it hat (1 Schritt ~ 1 Bit-Operation) und umgekehrt. Mit dem Schrnhage-Strassen-Algorithmus erhMt man so einen 0 (n log n log log n)-Algorithmus zum Berechnen der Quadratwurzel. 0 0 k A log Denotations the set of natural numbers {1, 2 .... } ,. f (n) f=0 (g) for two functions f, g : N ~ N means that nm sup-< oo .~ g (n) f-0 (g) means that f = 0 (g) and g = 0 (f) for x > 0, Lxa = max {y e N w {0} ] y_< x} logarithm to the base 2

This paper presents a trajectory for teaching multiplication of integers using the context of assets and loans. This context provides an experiential basis to understand all the sign rules, including 'minus times minus' without having to resort to distributive property. The paper also extends the argument for making a distinction between operation and number to multiplication of integers, in order to support sense-making by children.

arXiv (Cornell University), 2016

In this work a rationalized algorithm for calculating the quotient of two complex numbers is presented which reduces the number of underlying real multiplications. The performing of a complex number division using the naive method takes 4 multiplications, 3 additions, 2 squarings and 2 divisions of real numbers while the proposed algorithm can compute the same result in only 3 multiplications (or multipliers-in hardware implementation case), 6 additions, 2 squarings and 2 divisions of real numbers.