Category Archives: arithmetic

Matrix inversion, row ops, program added

The row operation matrix inversion method is so neat and ingenious, and it has the same operations for all dimensions of matrix.

Here is a step by step approach, where firstly the dimension is chosen, then the first of the buttons is selected (Start). After which the buttons are selected in order. The states of the left and right matrices are displayed at each stage, and finally the identity matrix appears on the left, and the inverse of the original matrix appears on the right.

http://mathcomesalive.com/mathsite/matrix%20prog%203.html

matrix-inverse-screen-dump-0

The first display shows the original matrix on the left. Nothing has been done yet.

 

matrix-inverse-screen-dump

The second one is the 2 x 2 matrix inverted.

matrix-inverse-screen-dump-2

The third is the 4 x 4 matrix inverted.

The matrices can be altered with the file name for the application, “matrix prog 3.html”. using right click and “view page source”. You are then on your own, with javascript !!!!!!!!

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Filed under arithmetic, computer, discrete model, engineering, javascript, Uncategorized

Negative numbers, the minus sign, abstract algebra.

I was pondering the reality of negative numbers and after figuring out that a sequence of dots on a line can be extended in each of the two directions, and then arbitrarily selecting one dot as “the zero”. The line can be further labelled as 1, 2, 3, … to one side and -1, -2, -3, … on the other side.
(better to label the 1, 2, 3, … as +1, +2, +3, … and consider the lot as “signed numbers”)

Soon proceeding towards arithmetic I concluded that 7-3 is 4, and also 8-4 is 4, and therefore 13-9 is 4, and then 3-7 is -4, and -2-2 is -4. It was then observed that if a-b=c then a-y-(b-y) is also equal to c, regardless of the signs of the specific numbers involved.
This of course is stunningly obvious when looking at the signed difference of the first and the second number as an extended number line diagram.

The outcome of all this was an arithmetic for 0, 1, 2 modulo 3, and  the signed difference x-y is a binary operation diff(x,y) with table:

…x  … 0     1     2
y
0         0     1     2
1         2      0    1
2         1      2    0

Example: 1-2 is -1, which is 2 modulo 3

So a non abelian, non associative algebra with a not quite identity satisfies the conditions, where A=1, B=2 and C=0
—————————————————–
There are three objects and an operation called “doesn’t have a name”.
Two are similar, and the third is a bit different
They are paired to yield a single object as follows:

AA = BB = CC = C
AB = BC = CA = B
AC = CB = BA = A

Notice that BC and CB are different, so non-abelian.
Worse is that (AC)A = C and A(CA) = B are different, so non-associative.
And consequently A and B and C are different.
—————————————————-

Interestingly, and maybe separately, the minus sign behaves very differently from the plus sign:

a-(-b) is a+b, but there is no way of writing a-b using only addition.

This means that all expressions can be written with “minus” alone.

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Filed under abstract, algebra, arithmetic, Uncategorized

Scary, and not just mathematically speaking …

Try this for size:

https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff#.a7llwy3mv

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Adding fractions – phew!

Who needs LCM ?

First, three views of LCM with no comments :

1: Change them to equivalent fractions that will have equal
denominators. As the common denominator, choose the LCM of
the original denominators. Then the larger the numerator, the
larger the fraction.

2: Jun 26, 2011 – If b and d were same it was easy to find LCM
since if denominators are same, we just need to find LCM of
numerators, hence LCM of (a/b) and (c/b) would be LCM(a,c)/b.
So we have to first make denominators of both the fractions same.
Multiply numerator and denominator of first fraction by LCM
(b,d)/b.

3: The GCF and LCM are the underlying concepts for finding
equivalent fractions and adding and subtracting fractions, which
students will do later.

 

Now we can do fraction addition without LCM. It just needs the use of the distributive law, and the result shows the way in which the divisors combine.

fraction-addition-png-1

And now using 3/4

fraction-addition-png-2

But the best one is via multiplication ……

fraction-addition-png-3-easy

Now for multiplication and division.

fraction-multiplication-png

fraction-division-png

 

 

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A minus times a minus is a plus -Are you sure you know why?

What exactly are negative numbers?
A reference , from Wikipedia:
In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”.
He came to the conclusion that negative numbers were nonsensical.[25]

A minus times a minus is a plus
Two minuses make a plus
Dividing by a negative, especially a negative fraction !!!!
(10 – 2) x (7 – 3) = 10 x 7 – 2 x 7 + 10 x -3 + 2 x 3, really? How do we know?
Or we use “the area model”, or some hand waving with the number line.

It’s time for some clear thinking about this stuff.

Mathematically speaking, the only place that requires troublesome calculations with negative numbers is in algebra, either in evaluation or in rearrangement, but what about the real world ?
Where in the real world does one encounter negative x negative ?
I found two situations, in electricity and in mechanics:

1: “volts x amps = watts”, as it it popularly remembered really means “voltage drop x current flowing = power”
It is sensible to choose a measurement system (scale) for each of these so that a current flowing from a higher to a lower potential point is treated as positive, as is the voltage drop.

Part of simple circuit A———–[resistors etc in here]————–B
Choosing point A, at potential a, as the reference, and point B, at potential b, as the “other” point, then the potential drop from A to B is a – b
If b<a then a current flows from A to B, and its value is positive, just as a – b is positive
If b>a then a current flows from B to A, and its value is negative, just as a – b is negative

In each case the formula for power, voltage drop x current flowing = power, must yield an unsigned number, as negative power is a nonsense. Power is an “amount”.
So when dealing with reality minus times minus is plus (in this case nosign at all).

The mechanics example is about the formula “force times distance = work done”
You can fill in the details.

Now let’s do multiplication on the number line, or to be more precise, two number lines:
Draw two number lines, different directions, starting together at the zero. The scales do not have to be the same.
To multiply 2 by three (3 times 2):
1: Draw a line from the 1 on line A to the 2 on line B
2: Draw a line from the 3 on line A parallel to the first line.
3: It meets line B at the point 6
4: Done: 3 times 2 is 6
numberlines mult pospos
Number line A holds the multipliers, number line B holds the numbers being multiplied.

To multiply a negative number by a positive number we need a pair of signed number lines, crossing at their zero points.

So to multiply -2 by 3 (3 times -2) we do the same as above, but the number being multiplied is now -2, so 1 on line A is joined to -2 on line B

numberlines mult posneg
The diagram below is for -2 times 3. Wow, it ends in the same place.
numberlines mult posneg

Finally, and you can see where this is going, we do -2 times -3.

Join the 1 on line A to the -3 on line B, and then the parallel to this line passing through the -2 on line A:

numberlines mult negneg

and as hoped for, this line passes through the point 6 on the number line B.

Does this “prove” the general case? Only in the proverbial sense. The reason is that we do not have a proper definition of signed numbers. (There is one).

Incidentally, the numbering on the scales above is very poor. The positive numbers are NOT NOT NOT the same things as the unsigned numbers 1, 1.986, 234.5 etc

Each of them should have a + in front, but mathematicians are Lazy. More on this another day.

Problem for you: Show that (a-b)(c-d) = ac – bc – ad + bd without using anything to do with “negative numbers”

*******************************************

References.
Wikipedia:
Reference direction for current
Since the current in a wire or component can flow in either direction, when a variable I is defined to represent
that current, the direction representing positive current must be specified, usually by an arrow on the circuit
schematic diagram. This is called the reference direction of current I. If the current flows in the opposite
direction, the variable I has a negative value.

Yahoo Answers: Reference direction for potential difference
Best Answer: Potential difference can be negative. It depends on which direction you measure the voltage – e.g.
which way round you connect a voltmeter. (if this is the best answer, I hate to think of what the worst answer is)
********************************************

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Filed under algebra, arithmetic, definitions, education, geometrical, math, meaning, negative numbers, Number systems, operations, subtraction, teaching, Uncategorized

Plus ca change, plus c’est la meme chose

Apologies for no accents!

1: Khan Academy 2016 Subtraction to 1000 You don’t have to watch it all! Run it faster if possible.

2: Tom Lehrer (cover version) original from the 60’s The New Math

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What exactly is Base 10 arithmetic ?

Teacher: “Now we’re going to learn about base 10 arithmetic”.
Wise guy: “Is that where 3 + 4 = 12, or is it where 3 x 4 = 12 ?”.

I did a search on the net and found the term “base 10” all over the place. What does it mean?

An apparently annoying question:
“Does the 1 in 10 stand for the number 10’s in 10?”.

The interpretation of 10 in the system described as “Base 10” depends on the base of the system, so what is it? How do I find out?

We have here a logical problem. The term “Base 10” as a definition is self referential. It is more subtle than this definition of a straight line:

“A straight line is a line which is straight”.

The problem arises from the almost universal confusion between the two things:
1: The name of a number, in this case “ten” is supposedly implied
2: The symbols representing a number, in this case 10 in the base ten system”

So the answers to the questions “What is it? How do I find out?” above are “Unknown” and “You can’t”

Writing “Base 10” when you mean “Base ten” is probably the first step in making math meaningless.

 

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Filed under abstract, arithmetic, confusion, definitions, language in math, math, teaching

A. N. Whitehead on negative numbers (1911)

This is really worth reading. It is from his book, “Introduction to Mathematics”, published in 1911.

whitehead intro to math negative nos

 

 

 

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More on √2 – Common Crappiness Simply Seen (CCSS)

It’s strange how one can read something many times and miss the complete stupidity of it, in math at any rate.
This is from the CCSSM Grade 8:
The Number System 8.NS (Grade 8)
Know that there are numbers that are not rational, and approximate
them by rational numbers.
2. Use rational approximations of irrational numbers to compare the size
of irrational numbers, locate them approximately on a number line
diagram, and estimate the value of expressions (e.g., π2). For example,
by truncating the decimal expansion of √2, show that √2 is between 1 and
2, then between 1.4 and 1.5, and explain how to continue on to get better
approximations.

I need an approximation to √2. Just get me the decimal expansion, please. Oh, and I need it to 73 decimal places.

Do I have to explain to the authors of this garbage that the only way I am going to get anywhere with √2 is by a process of successive approximation, NOT THE OTHER WAY ROUND ! !

And just try doing this for pi.

“I know that there are irrational numbers”. “How do you know that?”. “Because my teacher told me”.

And where will I encounter π2 ? Or “estimate the value of pi-e”.

And when we get to High School we find:

Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational;
that the sum of a rational number and an irrational number is irrational;
and that the product of a nonzero rational number and an irrational
number is irrational.

I find real difficulties explaining the last point.

I am not proposing that we go as far as Cauchy Sequences or Dedekind cuts, but if they cannot do a better job than this the topic is best stopped at “√2 is irrational and here’s why”. How many students can prove that √2+√3 is irrational?

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Gross misuse of + and – and x and the one that’s not on my keyboard

Arithmetic is the art of processing numbers.
We have ADD, SUBTRACT, MULTPLY and DIVIDE
In ordinary language these words are verbs which have a direct object and an indirect object.

“Add the OIL to the EGG YOLKS one drop at a time”.
“To find the net return subtract the COSTS from the GROSS INCOME”.

In math things have got confused.
We can say “add 3 to 4″or we can say “add 3 and 4”.
We can say “multiply 3 by 4” or we can say “multiply 3 and 4”.
At least we don’t have that choice with subtract or divide.

The direct + indirect form actually means something with the words used,
but when I see “add 3 and 4” my little brain says “add to what?”.

There are perfectly good ways of saying “add, or multiply, 3 and 4” which do not force meanings and usages onto words that never asked for them.
“Find the sum of 3 and 4” and “Find the product of 3 and 4” are using the correct mathematical words, which have moved on from “add” and “multiply”, and incorporate the two commutative laws.

If we were to view operations with numbers as actions, so that an operation such as “add” has a number attached to it, eg “add 7”, then meaningful arithmetical statements can be made, like

“start with 3 and then add 5 and then add 8 and then subtract 4 and then add 1”

which with the introduction of the symbols “+” and “-“, used as in the statement above allows the symbolic expression 3+5+8-4+1 to have a completely unambiguous meaning. It uses the “evaluate from left to right” convention of algebra, and does not rely on any notion of “binary operation” or “properties of operations”.

If we want to view “+” as a binary operation, with two inputs then, yes, we can ascribe meaning to “3+4”, but not in horrors such as the following (found in the CCSSM document):

To add 2 + 6 + 4, the second two numbers can be added to make a ten,
so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

If + is a binary operation, which are the two inputs for the first occurrence of + and which are the inputs for the second occurrence of + ?
The combination of symbols 2 + 6 + 4 has NO MEANING in the world of binary operations.

See A. N. Whitehead in “Introduction to Mathematics” 1911.
here are the relevant pages:
whitehead numbers 1
whitehead numbers 2a
whitehead numbers 2b
whitehead numbers 3a
whitehead numbers 3b

And here are two more delights from the CCSSM document
subtract 10 – 8
add 3/10 + 4/100 = 34/100

In addition I would happily replace the term “algebraic thinking” in grades 1-5 by”muddled thinking”.

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Filed under arithmetic, big brother, Common Core, language in math, operations, subtraction, teaching