Mathematical Notation and Schools – 5

The Overloaded Equals Sign: Equivalence

In this series about mathematical notation and its impact on the learning of mathematics, the previous post started to look at the overloading of the equals sign – so many different meanings and uses cramped in this single notation “=”.   Sometimes, like the “=” key on a calculator, the meaning is “do this now”, perform the calculation and get me a single number.

Other times, the “=” sign is used to indicate equivalence between two different ways of describing what is (in some sense) the same thing.  If you compare $x + 1 + 1$ with $x + 2$, you may agree that they look different, but that they have something very important in common.  That is, by some measures they are different, by another measure they are the same.  This phenomenon is known as equivalence.  Equivalence is a very powerful and general idea.  It is different from equality, in that two equivalent things need not be the same.  They just need to match in the particular attribute that you care about.   In textbooks, as well as in actual practice, the thing you care about is most often left implicit.  When we say that $x + 1 + 1$ and $x + 2$ amount to the same thing, what we are focusing on is the values you get when the expression is evaluated for a particular choice of x, and then noticing that these values will be the same for both expressions regardless of what number we choose for x.  By focusing on the values, and on nothing else, we say that $x + 1 + 1$ and $x +2$ are the same.  And yet they are not the same in other aspects.   These other aspects may be considered trivial, but that doesn’t make them less real.  For example, many teachers would not accept $x + 1 + 1$ as a correct answer to a problem, saying that it should be simplified to $x + 2$.  This is typical for the realm of equivalence: some things are more equal than others.

The particular kind of equivalence where we focus on the values when choosing a particular value for x, this kind of equivalence is often called identity.  The standard mathematical notation for both equivalence and identity is “≡”, though ‘standard’ doesn’t mean universal.  Most people and textbooks write $x + 1 + 1 = x + 2$.  The idea is that both sides are equal, regardless of the value of x.  The part of “regardless of the value of x” is not made explicit.

The use of “=” for identity is often confused by students with the use of “=” for equations.  When we write $x + 1 = 5$ , we are not saying or suggesting that both sides are equal regardless of the value of x.  We’re saying almost exactly the opposite: we’re saying that by telling you that $x + 1 = 5$ , we’ve given you a powerful hint or clue as to what value x should have.  Students often rely on verbal clues like “simplify” or “expand” or  “evaluate” or “prove” or “solve” to figure out what they are supposed to do.   Teachers’ consistent use of “≡” for identity, even if the student is never asked to do the same, and even if the textbook doesn’t play along, is still a powerful tool for students to become aware of the different meanings for “=” they see in their textbooks.

Hence I propose that teachers use “≡” instead of “=” whenever the intended meaning is that the two sides are identical, that is, they evaluate to the same value whatever is chosen as the  value for x.  In my experience, both students and teachers find this helpful.

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Mathematical Notation and Schools – 4

The Overloaded Equals Sign: “Do This Now”

In this series about mathematical notation and its impact on the learning of mathematics, we’ve looked so far at the raised minus sign for negative numbers, the various ways of denoting multiplication, and various ways to denote the sequencing of operations.  In the next few posts, I’ll be looking at the different mathematical ideas that are expressed, in standard notation, through the equals sign “=”.

In this post, we’ll look at the most familiar use of the equals sign, the one learned first, and the one in common use on calculators.  In school, we all know what we are supposed to do with:

5 + 4 =

7 + 8 =

5 – 2 =

The job here, as we all learned, is to put a number to the right of the “=” sign, and this number is called “the answer”.  The number we put there is either the correct answer or the wrong answer, and it is the job of the teacher to decide which is which.  This is such an important part of the teacher’s job that the teacher even has a special edition of the textbook, different from the students’ version, and the teacher’s edition includes something called an “answer key”.  The answer key helps the teacher in this all-important task.  Like a traditional country doctor who is instantly recognizable by the white coat and the stethoscope, the teacher is distinguishable by the answer key.

Any well-trained kid will feel an urge, whenever she sees an equal sign with nothing to the right of it.  The same urge is felt, clearly, by a four-function calculator whenever the “=” key is pushed.  Something needs to be done!  Right now!  We need a number, and we need it stat!

This means that, at this stage, the equals sign has a clear direction to it, it is not symmetric in its use.  For a kid, seeing
5 + 4 = 9 would look normal: it would look like an addition problem that somebody else had already answered.  Yet
9 = 5 + 4 would look positively strange, and 5 + 4 = 6 + 3 would look positively wrong.  To adults these don’t look wrong, but that means we are used to the equals sign in different contexts, for other purposes.  In upcoming posts, I’ll investigate those other uses.

Staying with the “do this now” meaning of the equals sign, let’s take inventory of other ways to indicate this meaning, both in school and out.  From this, a reasonable proposal might emerge for notational schemes that teachers could use to help disambiguate the mathematical ideas for their students.

If you listen to how 5 +4 = 9 is pronounced in parts of the English speaking world: “five plus four makes nine”.  Though a minority pronunciation (I assume “five plus four equals nine” is considerably more common), the “makes” pronunciation connotes a production, an action with a direction and an intention.  As one very obvious way to indicate direction is by using an arrow, “→” (which we might pronounce as “makes”), let’s play with this notation and see how it looks.  In an earlier post, I showed
75 + 10 = _______  × 1.09 =  ______ + 4 = _____ as a way to show the sequence of operations in ((75+10)×1.09)+4, knowing that this use of the equals sign upsets many teachers.  So here is an opportunity to see if it looks better using the “→” instead of “=”:
75 + 10 → _______ × 1.09 → _____ + 4 → _____.   I do think it looks better, but that isn’t a strong enough argument for proposing the → symbol.  Other notations may look even better, and also look more natural.  Moreover, other notations may be better able to coexist with standard textbook conventions.   We might take inspiration from what computer languages have long used for assignment operators, where it was desirable to have a notation for assignment that was distinct from a notation for equality.  If you ever learned the Basic programming language, it may have looked strange that you saw constructs like X = X + 1 used there.  Many programming languages use the construct “x := x + 1” instead.  The symbol “:=” for assignment doesn’t deviate too far from the “=” sign but makes the directionality clear.  But for our purposes in disambiguating 8+4=, the assignment operator is oriented the wrong way around.

In light of this, I propose “=:” as an informal notation for “makes”:

7 + 5 =: 12

75 + 10 =: _______ × 1.09 =:  ______ + 4 =: _____

The “=:” is easy to use in a classroom, is easily rendered from a keyboard; the extra colon is relatively unobtrusive, and can gradually be dropped as students get older.  It is something the teacher can use to disambiguate the various meanings of “=” in the standard notation, and the teacher can do this without ever requiring the students to use this notation themselves.  The notation is easily explained to parents,  though little harm is done if such explanation never reaches the parents.

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Mathematical Notation and Schools – 3

Notations for Sequencing and Nesting (Parentheses)

In  this series, I’m exploring mathematical notation as it is used in schools, and how these notations help or hinder students’ understanding of mathematics.  In the first installment of this series, I spoke positively about the practice in US schools to write negative numbers with a raised minus sign.  In the second installment, I explored the use of an asterisk (*) as an  alternative to the multiplication sign starting in middle school or slightly before.

In this post, I want to explore the various ways that sequencing of operations is indicated in schools.  But to set the stage, let’s first take a look at how we indicate to normal adults what sequence of operations to perform.  Here is an example, from one of the first Google entries of the “order form” search:

In such an order form, certain information must be provided, like Item Number and Description, that indicates what kinds of things are being ordered.  Then there are the columns labeled Quantity, Price and Amount.  You will most likely recognize this as requiring the Amount, on a given row, to be determined from the Quantity and the Price by multiplication.  The row labeled Subtotal will then be obtained from the Amounts by adding them all up.  To this Subtotal, a Discount will be applied, presumably some fixed percentage of the Subtotal, or some sliding scale.  The row labeled Total will be obtained by subtracting the Discount amount from the Subtotal.  The row labeled Tax (CO) presumably is a known percentage of the Total; the row labeled Shipping may be a fixed amount or may be found from a table.  Last, the AMOUNT DUE is obtained by adding Total, Tax (CO) and Shipping amounts.  In this set up, a sequence of actions is implied, leading from Quantities to the AMOUNT DUE through the Prices of the Item Numbers.  It makes little sense to compute the TAX (CO) before the first Quantity has been multiplied by the first Price.  On the other hand, there is some latitude in what order the multiplications of Quantity and Price are done, as long as all or done before the Subtotal is attempted.

I suggest we’ve all seen variants of this order form, and thus are familiar with the implied sequencing of actions in it.  Note that this is communicated to us in a relatively standard way, and yet there is no parentheses nor My Dear Aunt Sally to be found.

In secondary school, we may encounter formulas that look like  ((75+10) × 1.09 ) + 4, and to make sense of this, we must understand parentheses and “order of operations”.  In elementary grades, it is not very common to see multiple operations combined in a single formula.  It wouldn’t be uncommon to see it represented in elementary grades as follows:

75 + 10 = _______  × 1.09 =  ______ + 4 = _____ ; and even though many teachers would balk at this (mis)use of the equals sign, students don’t seem to have any problem with the sequencing aspect.  They seem to get they are supposed to proceed from left to right:  75 + 10 = 175, then 175 × 1.09 = 190.75 and finally, 190.75 + 4 = 194.75.

Like the order form we showed, there are ways to indicate the sequence of arithmetic operations that don’t look at all like mathematical  formulas with parentheses.  In an earlier post, I worked with representations like the following:

which can be used equivalently to $( 12 x + 8 ) / 4$.  Note that each box works on its own input and produces an output in return.  The sequence of arithmetic operations is obvious from the way the boxes are connected.

None of these examples is intended to suggest that parentheses are bad, merely that there are many other ways to indicate in what sequence operations should be performed.  When you learned the quadratic formula, did you ever notice that it doesn’t have parentheses?  If you’ve forgotten: one of the solutions of the equation $a x^2 + bx + c = 0$ is $\frac{-b + \sqrt {b^2 - 4ac}}{2a}$.  Here, the length of the horizontal line of the division is used to indicate what belongs to the numerator; similarly, the length of the horizontal part of the square root sign is used to indicate what belongs ‘inside’ of the square root.

Now let’s look at the kinds of formulas where traditionally we might use parentheses:

In standard notation, the formula on the top would be written as $x^2 + 2x$; the bottom one would be written as $x (x+2)$.  In the picture, parts of the formula have been encapsulated in bags – like crude ellipses.  What the picture on top tries to suggest is that the formula $x^2 + 2x$ is basically a sum of two parts.  One part is $x^2$, the other part is $2x$.  Similarly, the picture on the bottom tries to suggest that the formula $x (x+2)$ is basically a product of two parts.  One part is $x$, the other part is $x + 2$.

We can think of parentheses as an in-line version of the ellipses used in the picture: a rendering of the left part and the right part of the ellipses while leaving out the ceiling part and the floor part.  And, indeed, the formulas could be written as $(x^2) + (2x)$ and $(x)(x+2)$, respectively.

From my observations with students in middle school, the notion of using parentheses is not that difficult to grasp.  More difficult for the students is getting facile with when these parentheses are left out.  In the standard notation, parentheses are only written “when needed”, by which is meant that parentheses are only written if leaving them out would alter the meaning of the formula.  In the standard system of mathematical notation, there is a fairly good reason for wanting to eliminate as many parentheses as possible from a formula: when you get too many parentheses, especially if they are nested, things become a bit confusing.  How easy is it to make sense of the following formula?

x = (-b + sqrt(b*b-4*a*c))/(2*a)

This is another rendition of a solution to the quadratic equation.  It has three left-parentheses: “(“, and three right-parenthesis: “)” – and each left-parenthesis is matched with a particular right-parentheses.  The matching is relatively easy to do with a pencil if the thing is printed on paper, but not quite so easy to do just in reading it.

The adult’s insistence on dropping all unneeded pairs of parentheses has an unexpected consequence for students: students start to think of parentheses as having something to do with multiplication.  They see $(x+1)(x+3)$ and are asked to simplify it and they expand it to $x^2 + 4x + 3$, but then if you ask them to simplify $(x+1) + (x+3)$, they likewise treat it as a multiplication instead of an addition.  The sight of the parentheses seems to make them think that multiplication is in order – after all, that’s where they have seen parentheses the most: in factors.  Sometimes teachers explicitly encourage students to think of the parentheses as indicating multiplication: they’ll tell their students that $(x+1) + (x+3)$ has a hidden multiplication by 1 in it, so that the formula really means $1(x+1) +1(x+3)$, which is then subject to the distributive property and expanded to $1*x+1*1 + 1*x +1*3$, which then becomes $1x + 1 + 1x + 3$ and finally $2x + 4$.

My proposal is to introduce the bags early and frequently as ways to annotate a formula and indicate the order in which things happen.  Bags can be nested, meaning you can have bags inside of bags.  This use of bags, as in the picture below, can start in elementary grades as multiple operations are introduced.

Soon after, ‘real’ parentheses can be introduced as an abbreviation for drawing the bags, as follows:

After a while, kids can imagine the whole bags, just from seeing the parentheses.  The teacher can draw bags or parentheses interchangeably, while noting that the textbook only uses parentheses.

In middle school, I propose teachers use bags and parentheses whether they are “needed” or not, whenever their use can clarify the structure of a formula.  Teachers should use parentheses both for multiplication and addition, so students don’t automatically associate parentheses with multiplication as they tend to do today.

At the end of middle school, students can be eased into the standard notation, through being encouraged to write parentheses only when needed, like the textbooks already do.

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Mathematical Notation and Schools – 2

Notations For Multiplication

In the first part of this series, I looked at the practice in American schools to use a raised minus sign for negative numbers: e.g. ⁻7 for negative seven, in contrast to the “-” in 10 – 7.

Now I want to look at different notations for multiplication, and suggest one that is not currently widely used in schools but has wide currency elsewhere.

In elementary schools, by far the standard way to denote multiplication is by using the “×” sign.  We write 3 × 4 = 12, and we typically pronounce that as “three times four makes twelve”.  The main reason that “×” is so closely associated with elementary school is that this symbol is so easily confused with the letter “x”, and “x” is so much the domain of secondary school mathematics, as the primary symbol for an unknown amount.  At least, this is the standard explanation passed on from the distant past as to how come we switch in middle school to a different notation for multiplication.  This new notation is “•”, a dot.  So “a•b” is the new way to write “a times b”.  Unfortunately, this dot is not suitable as a universal symbol for multiplication either, since “3•4” (three times four) is very easily confused with “3.4” (three and four tenths).  In other words, the dot looks a lot like a decimal point, and neither your typical kids’ handwriting nor a typical laptop keyboard will let you easily distinguish the two.  So, the middle school textbooks tend to carefully use the “•” for multiplying letters (a•b), but also combinations like 2•a, but then switch to the old “×” to render “3×4”.

At roughly the same time, or later, the textbook will also introduce the idea that multiplication need not be written at all:
“ab” is read as “a times b”.   This very useful convention, which has been standard for centuries now, allows us to write
$2x^2 + 6x + 4$ to mean $2 \times x \times x + 6 \times x + 4$, eliminating any confusion between × and x.

However, this notational standard of writing nothing for multiplication (more precisely, to indicate multiplication by juxtaposition) comes at a cost.  This cost is quite considerable.  Yet because we adults are so used to this standard, we are barely aware of the cost.  To become aware of part of this cost, you can watch kids when they are first learning this convention.  Since juxtaposition doesn’t work for 12 times 34: nobody would see 1234 and be able to tell whether it meant 1234 or 123×4 or 12×34 or 1×234, a different convention for multiplying numbers is needed.  Teachers will tell kids to write 12(34) to mean multiplication.  The multiplication is still expressed through juxtaposition, and the parentheses are now used to make clear what the pieces are that are being juxtaposed.  For kids, this is not real obvious, so here is one cost.  A bigger cost, in my mind, is that it pretty much forces variables to be single letters.  This is not something that kids seem to have trouble with, and not something that teachers typically see as a limitation.  In the standard notation, “cost” means “c×o×s×t”, and can’t be a single variable that indicates how much something costs.  And indeed, if you look at college-level math texts, you will see how many alphabets mathematicians have appropriated (Greek, Hebrew, German) in order to have enough different symbols.  Even so, the convention is not used consistently.  When students see sin(45) in secondary school, they know somehow that this means the sine of an angle of 45 degrees, and not s×i×n×45.

When communicating mathematical formulas to computers, whether when writing computer software or entering formulas in spreadsheets like Excel, entirely different conventions are used, born from the limitations of typical keyboards.  The computer-based conventions resemble $\sin(x)$  a lot more than $\sum_{i=1}^{10} t_i$.
Since computers require very precise communication, a lot of work has been done to come up with clear and unambiguous languages for formulas.  Though there isn’t a single standard for these, there are commonalities in the conventions that have been developed.  In almost all of them, a variable is not restricted to a single letter, it can be a construct like total_cost_before_taxes.  In almost all of them, juxtaposition is avoided and all operations have to be explicit.  Many of these computer-based conventions rely on the “*” sign for multiplication.  This asterisk is found on all keyboards, and looks somewhat like a cross, a “×”.  Most importantly, it is not easily confused with x.

Here, then, is my proposal: introduce the asterisk “*” fairly early on in school as a variant of “×”, right around the same time that decimal numbers come into play.  Don’t make a big deal of it, don’t require its use, just get students used to seeing it as meaning the same thing as “×”.  This is not a new thing to students: variant notations for division are very common: “÷”, “⁄”, “—”.  So now multiplication has a variant, and this variant can gradually replace “×” and instead become a variant of the juxtaposition learned in middle school.  So a×b moves through a*b to ab, whereas 3×4 moves to 3*4 and never need become 3•4 or 3(4).

Pros: no confusion, no ambiguity, easy to use on a keyboard.  Kids learn this easily ( I’ve used this for years, and never had to explain it to kids beyond “this is just another way of writing ×”).   Secondary pros: easy transition to Excel or C# or a number of other computer languages that use this same convention for multiplication.

Cons: teachers are generally unfamiliar with this convention, and especially elementary grade teachers.

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Mathematical Notation and Schools: The Series

Here’s a summary of the series (thus far) of posts on mathematical notation, with links. This allows you to read them in order, from oldest to most recent.

In this series, I’m investigating variations of standard mathematical notation, with an eye on its use in schools.

1 – In the first installment, I look at the raised minus sign you see in American schools, to indicate a negative number.  You see ⁻7 to indicate negative seven, distinct from the normal minus sign used in e.g. 10 – 7 to indicate subtraction.  How is that useful, and if so, why do people stop using it after a certain grade?

2 – In the second installment, I look at notations for multiplication, and the way that “×” tends to be phased out in secondary school.  You see “•” used in middle school, but both “×” and “•” are too easily confused with common symbols in middle school: “x”, the variable, and “.”, the decimal point.  The secondary school (and later) convention of simply juxtaposing things to indicate multiplication (where 4ac means 4 times a times c) works, but makes it necessary to write 3(4) to mean 3×4.  I suggest an alternative in using “*” as a variant for “×” from about fifth grade on.

3 – In the third installment, I look at notations for sequencing and nesting of operations, and suggest an alternative notation for parentheses which I call bags.  These bags are easy to draw, but hard to type on a keyboard.  On a keyboard, these bags naturally devolve into ordinary parentheses.  Bags are easy to nest, and they look just as natural for terms as they do for factors.

4 – In the fourth installment, I started a look at the many uses of the equals sign, confusing if not conflicting, and looked at a common non-symmetric use of it, the one you see on many calculators: “do this now”.  What alternative notations might help disambiguate this?

5 – In the fifth installment, the use of “=” for equivalence, e.g. $x + 1 + 1 = x + 2$ is highlighted.  The “=” is used here to assert that left side and right side will evaluate to the same value regardless of the value chosen for x.  Is this use of “=” worth differentiating from the use of “=” in equations, and if so, what notation would help student learning?

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Mathematical Notation and Schools – 1

Notations for negative numbers

This blog entry marks the start of a series about mathematical notation and its uses in school.  I intend to look in detail at aspects of the standard mathematical notations, and how these help or hinder understanding of math in the various grades.  I also intend to look at special notations introduced in school to aid in understanding key math concepts, and to propose additional such notations.

A simple and useful example is the use of two different “minus” signs in school (at least in the USA), the normal “-” sign for subtraction, and the raised minus sign for indicating a negative number, ⁻7 for negative seven.  This notation, typically in use up to 8th grade, allows the text books to write  8 – ⁻7 rather than having to write 8 – (-7), for instance.  Teachers typically think of this negative number notation as a kind of training wheels.  This is akin to the use of pointing in Hebrew to indicate vowels, which is primarily used in texts for children or beginning Hebrew students.  Indeed, most students are encouraged to drop this special notation as they get further along in school.

From my own observations with students up to 8th grade,  this use of a distinct minus sign to mark a negative number is both successful and non-intrusive.  By ‘non-intrusive’ I mean that, as far as I can tell, the notation can usefully coexist with the standard notation, and that if some 8th grader decided to keep using that notation for the rest of his or her life, the rest of the world would barely notice and would not be in the least inconvenienced.  This would be true even if the student ended up in an engineering school or became a math major.  The use of the raised minus sign (the official name of this symbol appears to be superscript minus) is a harmless variation out there in the world, and is demonstrably useful in the lower grades.  These may be precisely the conditions under which a variant notation can spread far and wide.

In this series, I will bring up other examples of variant notations and try them out, to see if they meet these same tests: harmless in the wider world, and useful in the context of math education.  In addition to looking at notation, I will also occasionally bring in examples of variant vocabulary and terminology.

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Operators, Functions, and Properties – part 43

In the last several posts in this series, we’ve looked at cyclical patterns of behavior of numbers and operators.  For example, in the previous post we saw that when counting up, the last digit of a number consistently repeats after ten counts.  We also saw that the last digit of a sum only depends on the last digit of the (positive whole) numbers we are adding.

This is shown in the table above, and  it is easy to check this on particular numbers, e.g. 28+2076.  The table predicts that the last digit of 28+2076 is the same as that of 8+6, which the table shows us as 4.  Since 28+2076=2104, its last digit indeed matches the 4 from the table.

If we wrap a number line around a cylinder, kind of like a slinky, as we did in previous posts, we can get cyclical behavior that is not necessarily based on cycles of ten.  Cycles of ten just happen to be very easy to spot from looking at the digits of the number, as our standard way of writing numbers is based on cycles of ten.

To see cycles other than cycles of ten, we need to stop focusing on the last digit of the number, and instead focus on the remainder after division.  This actually works for cycles of ten also, so let’s look at that first.  What is the remainder of 7653 after division by 10?  Well, 7650 is the multiple of 10 that comes closest to 7653 (yet stays below it), and so there are 3 left.  The remainder after division by 10 of a whole positive number is simple its last digit.

The diagram above shows the same result we had before, but this time stated in terms of remainders.  Now we can pursue similar results using different cycles.  Below is one based on remainders after division by 9:

This result is the basis of the well-known procedure of “casting out nines” by which people used to check their arithmetic before the widespread availability of electronic calculators.  The “casting out nines” process was considered convenient in part because the remainder after division by 9 is relatively easy to determine.  Though not as easy as finding the remainder after division by 10, you can get the remainder after division by 9 without doing long division, or any division for that matter.  Instead, it is sufficient to add up all the digits of the number.  For the number 7653, adding up all the digits gets us 21.  The claim (not further pursued here) is that the remainder of 7653 after dividing by 9 is the same as the remainder of 21 after dividing by 9.  We can do this trick – adding all the digits – repeatedly until we arrive at a single digit.  For 7653, this takes two steps: 7653 → 21 → 3.  And indeed, 7650 is a multiple of 9 (9×850), so we would agree that the remainder for 7653 is 3.

For the rest of this post, I’d like to introduce some of the common mathematical notation for expressing the kind of relationships we’ve been exploring.  It gets too cumbersome to continue to say things like “the remainder after dividing by 9”.  Unfortunately, the standard notation, though compact, has some drawbacks as well.  But here we go:

7653 ≡ 21 (mod 9)

is the standard notation for: “both 7653 and 21 give us the same remainder after we divide each by 9.”

The standard pronunciation of this notation would be: “7653 is equivalent to 21 modul0 9″.

In textbooks you often see it defined as follows:

a ≡ b (mod m) if a−b is a multiple of m.

Usually, this notation is only used if m is a whole number, and positive.  Typically, a and b are whole numbers as well, though they can be negative.  For example,

−14  ≡  6 (mod 10)

is considered valid, as -14 and 6 are 20 apart, an exact multiple of 10.

Many people think of the modulo notation as having less to do with division than simply with repeated subtraction or addition.  You can get from -14 to 6 by repeated addition of 10, and you can get from 7653 to 21 by repeated subtraction of 9.

Operators, Functions, and Properties – part 42

In this series, we’ve been looking at black boxes, where you put something in and something comes out.  From looking at its behavior, you can often propose a model for what is happening inside.  If you build the model, and it behaves the same way as the original black box, you’ve got something that’s equivalent to the original black box.  Equivalence doesn’t mean equality – there may be many ways to model what is going on in the black box, and there may be no easy way to choose one model over another.  Our exploration in this series has led us in many interesting directions.  Most recently we looked at what happens if you take a number line and bend it, like a slinky, into a cylindrical shape.  In this post, we bent the number line so that numbers 4 apart would line up one above the other, and each number on the number line could be associated with either north, east, south or west.  In the most recent post, we imagined the number line being wrapped around an old-fashioned clock so that each number would correspond to an hour from 1 to 12.  On these slinky number lines, we looked at the effect of one operation, addition, and found that whether the result of addition of two numbers points north, east, south or west depends only on the compass direction of the two numbers involved and on nothing else.

This repetitive nature of a number line bent around a slinky might also remind us of the scenario we looked at in the very first post of this series: there we looked at a mysterious machine with a “next” button and a window in which a number shows.  Each time we hit the button, another number shows in the window, in what seems to be a predictable repetition of the digits 3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,…   With the perspective of the recent posts, we can now think of a number line bent like a slinky – completing a full turn every ten numbers.  The mysterious box may move us along the number line on the slinky and show on the display where in the cycle we are.

The notion of things repeating after ten steps is of course central to our decimal number system.  It applies most literally to the right-most digit in a number, as we count up.  Let’s assume we start with the number 966839, as on the mechanical counter below, and then repeatedly count up by one.

We would produce the following sequence of numbers: 966839, 966840, 966841, 966842, 966843, 966844, 966845, 966846, 966847, 966848, 966849, 966850, 966851, 966852, 966853, …  When you only pay attention to the right-most digit, you see them repeat in a cycle of ten steps.  But the other digits in the number each cycle through a set of ten values in turn.  It just happens a lot slower.  If you take the same set of numbers, and look at the second digit from the right, you would see the sequence 3, 4…4, 5…5, etc., where each digit sticks around for ten steps and then changes to its next value.  In the mechanical counter, this is realized by having each digit position consist of an identical wheel (ten digit values in the same sequence), and kicking each wheel forward only when the wheel to the right of it transitions from a “9” to a “0”.  The effect of this arrangement is that the wheels to the left rotate at a much slower rate than those on the right.  By putting enough wheels together in this arrangement, we can achieve very large cycles; in the mechanical counter shown, the overall cycle is one million.  This means that one million kicks of the right-most wheel will bring the whole counter back to its previous state, after which the whole thing repeats.

The rightmost wheel corresponds to the rightmost digit of the number.  If I have two numbers, and I focus exclusively on the rightmost digits, I can predict what the rightmost digit of the sum will be.  Also what the rightmost digit of the difference will be, and the rightmost digit of the product.  E.g. if I have a number X7 and another Y8, where the X and Y represent the digits that are covered up, you can predict that the sum will look like  Z5, and that the difference of X7 – Y8 will havea rightmost digit of 9, and the product X7 × Y8 will end in a 6.  Unfortunately, you can’t predict with certainty what the rightmost digit of the quotient is, even if there is no remainder.  This turns out to have something to do with 10 not being a prime number.

If it seems obvious to you that the rightmost digits of the sum and product only depend on the rightmost digits of the original numbers, please note that you cannot predict the next digit over quite so easily.  The second digit from the right of the sum does not depend only on the second digits from the right of the addends, and the second digit from the right of the product does not depend only on the second digits from the right of the factors.  However, it turns out that you can predict the final two digits of the sum and product by knowing only the final two digits of the addends or factors.  If our gear machine had wheels that had a hundred entries on them instead of ten, marked from “00” to “99”, we could again say that the position of the final wheel of the product could be determined from the position of the final wheels of the factors.

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Operators, Functions, and Properties – part 41

In this series, we’ve pictured operators as black boxes with inputs and outputs.  In the previous post we looked at rather ordinary addition from a less ordinary perspective: we imagined the number line arranged as a cylindrical spiral, like a slinky.  We imagined that numbers four apart would end up vertically aligned, with one above the other.

We’ll be looking at a variation of that in the current post, and our vehicle for introducing this will be an old-fashioned alarm clock:

We may be getting close to an era where children  won’t have any idea how to read time on a clock like this, never even having seen one in grandma’s house.  The saying “Even a broken clock shows the correct time twice a day”  will not make any sense to them either.  If a digital clock somehow got stuck and showed a single static display, we’d still expect it to show whether it was AM or PM, we would expect it to show the date, we might expect it to show the day of the week.  Even without the day of the week, such a clock would show the correct time only once a year.

The hour pointer (“Mickey’s little hand” for those from the right country and the right era) in the picture is pointing at the “1”, and exactly one hour later, it will be pointing at the “2”.  Each hour it advances one number around the clock face, and it keeps doing so: one hour after it is pointing to the “12”, it will point at the “1” again.  So if we focus only on the hour hand, and ignore anything else about this clock, we would expect to see behavior that strictly repeats every 12 hours.

The picture above symbolizes the “advance one hour” or “one hour later” effect on the position of the hour hand.  So if the input of the box s “11”, the output will be “12”, if the input of the box is “12”, the output will be “1”.  This is shown below in table form:

Now please look at the diagram below:

The top row of this diagram shows the repeated application of the “one hour later” operator.  The input on the far left will be a clock time (whole hours only), and as we move rightwards through the boxes, we trace the time on the clock going forward one hour per box.  The bottom row is a simple counter, starting at zero and going up by one each time.  What’s different from what we’ve done before: the intention is that the two run together: at one point, both machines would be at A, and some time later, both would be at B – they stay in synch.  There are other ways to model the synchronization of different parts of a machine, and we’ve done so earlier in this series.

At A, the bottom machine shows an output of 1, and the top machine shows the time one hour later than the input time.  At B, the bottom machine shows an output of , and the top machine shows the time two hours later than the input time.  Similarly, at the very right of the diagram shown above, the output of the top row will show the time 14 hours later than the input time, and the output of the bottom row will simply show 14.

In the diagram above, we show the same idea, two machines running in synch, one tracking the hours on the clock face, the bottom one counting the hours gone by.  We no longer assume the bottom row starts with zero.  Note that this time we’ve shown exactly 12 stages.  For the bottom row, looking at the net effect of running 12 stages – of adding 1 in each stage – is easy.  The next effect is adding 12 to the incoming cont.  For the top row, the net effect of running 12 stages – of advancing the clock face one hour in each stage – isn’t that hard either: regardless of the position on the clock face when we start, after 12 stages we will have made a full cycle around the clock face, and we will have returned exactly to the point where we started.  We can show this as follows:

The top box is what we’ve called an empty box or an identity operator in earlier posts.  One of the implications of this is that if we a long train of the “one hour later” operators, we can take any sequence of 12 consecutive “one hour later” operators, and remove it from the train without altering the net effect of the train.

So if we had a machine with 38 stages, with each stage a pair of “one hour later” on top and an ordinary +1 on the bottom, we could find 12 consecutive stages and reduce them as above, and repeat this process till we were left with:

From the bottom row we can confirm that there were indeed 38 stages (the net effect of the bottom  row is +38); from the top row it is clear that the net effect is two stages’ worth of “one hour later” (i.e. two hours later).

In terms of their effect on the clock face, 38 stages and 2 stages are equivalent.  The standard language for that is that 38 and 2 are equivalent modulo 12.

We will look further into this in subsequent posts.

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Operators, Functions, and Properties – part 40

In this series, we’ve spent considerable space playing with operators, which we’ve pictured as black boxes with inputs and outputs, often as part of a machine with state.  In more recent posts, e.g. this one , we’ve looked at operators that we likened to moving around on a contour line, towards a representative value in an equivalence class.

In the prior post, we looked at a number line on a slinky, and picked an equivalence relationship simply based on which numbers are vertically aligned.  We’re going to take a closer look at slinky arithmetic,

mapping numbers onto the slinky as below – so that 5 ends up directly above 1, and 6 directly above 2, etc.:

It may be helpful to think as the “1” position as South, the “2” position as East, the “3” position as North and the “4” position as West.  As you move around the spiral, you keep coming back to South etc, but you’ll be a little higher up.

Now let’s look at arithmetic on the slinky, starting with addition.   We can take some well-known addition fact, like
5 + 10 = 15, and then note the compass positions of the numbers involved.  5 is South, 10 is East, and 15 is North.  The question I’m now curious about is: If I start with a number in the South and add a number in the East, will I always end up with a sum in the North? After all, the compass directions seem to have nothing to do with how high up we are.  On this number line, East comes after South, and North comes after East, and West comes after North, and after West comes South again.

In the diagram above, the top line shows the compass marks for the numbers from 0-18.  The diagram at the bottom left shows  a normal addition table.  In the table on the right, all the numbers from the left have been replaced by their compass markings.  As you can see, the pattern on the right is completely regular and repeats after 4 rows, after 4 columns and after 4 diagonals.

From that table on the right, we extract a four-by-four core piece, and show it below.  It is this piece that repeats over and over.  We can translate the compass marks back into numbers, as shown below on the right.  We used the numbers 0-3 as representatives of the equivalence classes – unlike the situation in the prior post, we now have at least some reason to pick this set of numbers over other numbers.

In some regards, the table on the right does look like an addition table: most of the entries match what we would expect.  It is only where we would normally expect a sum of 4 or greater that the table looks somewhat strange.  Instead of 4, we find 0; instead of 5, we find 1, and instead of 6, we find 2.  This is at least consistent with the equivalence relationship we’re looking at here: 4 is directly above 0, 5 is directly above 1, and 6 is directly above 2 in our slinky.

The style of addition in this last table has a fancy name: it is known as addition modulo 4.  The same phrase, modulo 4, is used to talk about equivalence classes for whole numbers: when any two numbers that are 4 apart are considered equivalent.  For example, people would say that 5 and 9 are equivalent modulo 4, and that 9 and 13 are equivalent modulo 4.  By extension, 5 and 13 are also equivalent modulo 4.  Some people would describe equivalence modulo 4 in terms of division: two numbers are equivalent modulo 4 if they have the same remainder after dividing by 4.  Yet another way to talk about it is in terms of multiples: two numbers are equivalent modulo 4 if they differ by a multiple of 4.  All these ways of talking about it amount to the same thing, but that may not be immediately obvious.  “Two numbers are equivalent modulo 4 if they point in the same compass direction on the slinky” also expresses the same idea.

So far, we’ve talked about modulo 4 and only looked at the operation of addition modulo 4.  We can extend in both directions: we could look at subtraction, multiplication, and division modulo 4.  We could also extend in a different way and look at equivalence modulo 5 or modulo 6 etc.  We will take this up in future posts.

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