# Category Archives: Dr. Kenneth Iverson

## Signed numbers, Negative numbers

Negative numbers create problems for many math learners, and really, for good reasons.  We don’t really encounter negative numbers in real life that often.  If we somehow  bounce a check,  buy something that declines in value, or suffer a loss … Continue reading

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## Simple Division and Rational Numbers aka Fractions

Let’s return to simpler math for this post;  middle school math to be exact.    J has some unique features that make it an excellent tool for teaching division, fractions, and decimals. Most calculators and computer programs have historically made … Continue reading

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## Much Simpler Matrix Operations. Addition, Subtraction, and Scalar Multiplication.

Multiplying a matrix by a scalar (a single number) is a much simpler matter than multiplying a matrix by a matrix.  In the example below you just multiply each number in the matrix times 3:           … Continue reading

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## Determinants – The 3×3 Matrix.

Once again, we’ll take a problem from Kelley’s Algebra pg. 122-123, Problem 6.  This is a 3 x 3 Matrix, still a square matrix but the calculation of the determinants is not as straight forward as the 2 x 2 Matrix.  Both … Continue reading

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## Determinants – How to Calculate the Determinants for a 2×2 Matrix

In order to calculate the determinant for a matrix, it must be a square matrix, with equal rows and columns.  The easiest square matrix to find the determinant is one that is 2×2.   See Kelley’s Algebra pg. 121-122.  Below, … Continue reading

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## Matrix Multiplication Algebra

I wouldn’t wish pencil and paper matrix multiplication algebra on my worst enemy.  Well, on second thought……  Anyway, Michael Kelley describes it as, “The trickiest thing you’ll need to do with matrices.”  In Kelley’s Algebra, Chapter 9, pg 118-120, he … Continue reading

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## Skip Cramer’s Rule and just do Helzer’s Matrix Divide in J

In the last post,  we demonstrated the use of Cramer’s Rule as explained by Michael Kelley in Algebra Chapter 9.  We saw that Cramer’s Rule works in J, but in this post and in a previous post using  Matrix Divide … Continue reading

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## Matrix division and Cramer’s Rule

In this post, we will demonstrate how you can apply Cramer’s Rule in J,  using  a sample problem in Michael Kelley’s  in his Algebra, pg. 126  Chapter 9. The original system of equations:                … Continue reading

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## Matrix Division made simple

The following might be a somewhat advanced topic for some of you.  But, don’t be intimidated, I actually think you’ll get it.   Gary Helzer wrote An Encyclopedia of APL for I-APL Limited,  which was part of British APL.  Dr. … Continue reading

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## The Shape Command in J with Squares and Roots

Previously, I mentioned that W. Michael Kelley in his Algebra Book suggested you should memorize the first fifteen perfect squares.  Then, in his Pre-Algebra book on pg 278, he suggests you memorize the first twenty square roots.  I’ve never really … Continue reading

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