Tag Archives: arithmetic

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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Filed under algebra, arithmetic, fractions, Uncategorized

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

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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Filed under abstract, arithmetic, Number systems, teaching, Uncategorized

The square root of minus one asked me “Do I exist?”

Complex number.
“complex” as opposed to “simple” ?
“number” for what ?
Not for counting !
Not for measuring ! We’ll see about that !
“Square root of -1”, maybe, if that means anything at all !

Who needs the “i” ? It’s not essential.
Here goes…..

They say that (a+ib)(p+iq) = ap – bq + (bp + aq)i
But only if i is the square root of -1.

Getting rid of the i
Let us put the a and the b in a+ib together in brackets, as (a,b), and call this “thing” a “pair”.
This gets rid of the (magic) i straightaway.

Let us define an operation * to combine pairs:
(a,b)*(p,q) = (ap-bq, bp+aq)
This is the “pair” version of the “multiplication of complex numbers”.

It’s more interesting to read this as “(a,b) is applied to (p,q)”, and even better if we think of (p,q) as a “variable” and “apply (a,b)” as a function.
Ok, so we will write (x,y) instead of (p,q), and then
(a,b)*(x,y) = (ax-by, bx+ay)
Let us call the output of the “apply (a,b)” function the pair (X,Y)
Then
X = ax-by
Y = bx+ay
Now we can see this as a transformation of points in the plane:
The function “apply (a,b)” sends the point (x,y) to the point (X,Y)

Looking at some simple points we see that
(1,0)*(x,y) = (x,y)….no change at all
(-1,0)*(x,y) = (-x,-y)…the “opposite” of (x,y),
so doing (-1,0)* again gets us back to no change at all.
(0,1)*(x,y) = (-y,x)….which you may recognize as a rotation through 90 deg.
and doing (0,1)* again we get
(0,1)*(0,1)*(x,y) = (0,1)*(-y,x) = (-x,-y)….a rotation through 180 deg.

So with a bit of faith we can see that (0,1)*(0,1) is the same as (-1,0), and also that (-1,0)*(-1,0) = (1,0)…check it!
Consequently we have a system in which there are three interesting operations:
(1,0)* has no effect, it is like multiplying by 1
(-1,0)* makes every thing negative, it is like multiplying by -1, and
(0,1)*(0,1)* has the same effect as (-1,0)*

So we have found something which behaves like the square root of -1, and it is expressed as a pair of ordinary numbers.
It is then quite reasonable to give the name “i” to this “thing”, and use “i squared = -1”.

And generally, a complex number can be seen as a pair of normal (real) numbers, and bye-bye the magic !

When you think about it a fraction also needs two numbers to describe it.

Next post : matrix representation of “apply (a,b) to (x,y)”.

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The mathematics of harmonic means and the beat frequency

Two bits of math associated with the harmonic scale and the beat frequencies. See recent posts on music and math for where this came from.

1. Why is the word “harmonic” used for the “harmonic” series 1+1/2+1/3+1/4+…?
This is easy, the fractions in the series are exactly those proportions of a stretched string that yield the harmonics of the open string.
Why is the word “harmonic” used in the definition of the “harmonic” mean?
This looks obvious after the event, but I was stupidly surprised by it.
Let p and q be two frequencies, and a and b the corresponding periods.
Then a = 1/p and b = 1/q
The harmonic mean k of p and q is 2/(1/p + 1/q), which is 2/(a + b)
Call the period corresponding to this frequency h
Then h = 1/k = (a + b)/2
So the harmonic mean of two frequencies corresponds to the arithmetic mean of the two corresponding periods.
You can check to see that it works both ways !
So it’s all about sounds, music and harmony.

2. The formula for the beat frequency for two notes with frequencies f and g.
Beat frequency = |f – g|
But why ? This is TOO simple !!!
I found this by looking at the plots and making a table, then confirmed it by a quick visit to the internet, but no proof.
Here is one of the plots
beatpic7a
We are adding two sine waves together, so back to school math and always having to work the sine and cosine formulae out from scratch I got to sin(A + B) + sin(A – B) = 2sin(A)cos(B).
After turning this into the sin(P) + sin(Q) form and getting nowhere I realised that the first version held the answer.
Take A + B for the higher frequency f and A – B for the lower one g, so we have A + B = f(2πt) and A – B = g(2πt),
Solving we get A = ((f+g)/2) x (2πt) and B = ((f-g)/2) x (2πt).

So the combined signal has an amplitude of 2, a fast wave from the ‘A sine term and modified by a slow wave from the (f-g)/2 cosine term.
The slow cosine wave has a frequency of half the difference between f and g, but in each cycle of the cosine there are two pulses perceived as volume change, so the frequency of these pulses is twice as big, which is the difference between f and g.

We have ended up with a sine wave of frequency (f+g)/2 modulated by a cosine wave of frequency (f-g)/2. In radio communication this is called AM, or amplitude modulation (as opposed to FM).

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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

Subtraction and the “standard” algorithm

CCSSM talks about “the standard algorithm” but doesn’t define it – Oh, how naughty, done on purpose I suspect, since there are varieties even of the  “American Standard Algorithm”. Besides, if it is not defined it cannot be tested (one hopes!). I checked some internet teaching stuff on it, and as presented it won’t work on for example 403 – 227 without modification.

Anyway, I was thinking about subtraction the other day (really, have you nothing better to think about?), and concluded that subtraction is easiest if the first number ends in all 9’s or the second number ends in all 0’s. So, fix it then, I thought, change the problem, and here are the results

.Two simple algorithms for subtraction

I am quite sure that some of you can extract the general rule in each case, and see that it works the same in all positions.

While I am going on about this I would like an answer to the following-

“If I understand subtraction, and can explain the ideas to another, and I learn the standard algorithm and how to apply it, and I have faith in it based on experience, WHY THE HELL DO I HAVE TO BE ABLE TO EXPLAIN IT?”

I guess this post counts as a rant!

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A fun math/computing problem

I found this on http://www.playwithyourmath.com/ and adapted it a little.

The number 25 can be broken up in many ways, like 1+4+4+7+9

Let’s multiply the parts together,  getting 504 (or something near)

Problem 1: Find the break-up which gives the max product of the parts. 1+1+1+…+1 is not much use.

Problem 2: Find a rule for doing this for any whole number.

Problem 3: Put this rule in the form of a computer algorithm (pseudocode is OK)

Problem 4: Write the rule as a single calculation (formula)

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Egyptian fractions

I found this on Quora. What would the standard algorithm be, I wonder.
……………………………………………………………………………………………………
David JoyceDavid Joyce, Professor of Mathematics at Clark Uni… (more)

Suppose you have five loaves of bread and you want to divide them evenly among seven people.  You could cut the five loaves in thirds, then you’d have 15 thirds.  Give two of them to each of the seven people.  You’ll have one third of a loaf left.  Cut it into seven equal slices and give one to each person.

\frac57=\frac13+\frac13+\frac1{21}
There may be other solutions.   a = b = 3, c = 21.   (Egyptian Fractions)

……………………………………………………………………………………………………

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Fractional doggerel – verse problem

Mary’s mother brought a pizza
For her little kiddies, two.
“Johnny, you can have threequarters.
Mary, just a half will do.”.

Then the kiddies started eating.
Soon Mary grabbed her final piece.
“That’s mine” screamed Johnny, then the fighting
Broke the tranquil mealtime peace.

How much pizza then was eaten?
How much pizza on the floor?
Mother swore and left the building.
“I should have ordered just one more”.

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Filed under arithmetic, fractions, humor, language in math, verse