The Reflectionary

This week, liturgy resources based on Isa 55 and Psalm 63, and a somewhat longer than usual reflection. Probably best to get a cup of tea and a biccy before you start. (The TL;DR version – every time I think I know, I find there’s a whole lot more to know. It applies to God as well as numbers.)

Lent 3

Isaiah 55:1-9, Psalm 63:1-8

Bidding

Come, you who have no money. Come, buy and eat.
Come, you who thirst. Come to the waters and drink.
Come, you who seek the Lord. Come, find him while he is near.

Confession and Absolution

O God, you are my God and I seek you,
my soul thirsts for you, as in a dry and weary land.

O God, I confess that my thoughts have been far from you,
that my mind has focussed on the worries of the day,
and I have forgotten you.
O God, you are my God and I seek you,
my soul thirsts for you, as in a dry and weary land.

O God, I confess that my tongue has spoken words
that I now wish unspoken,
and I have dishonoured you.
O God, you are my God and I seek you,
my soul thirsts for you, as in a dry and weary land.

O God, I confess that my hands have hurt instead of helped,
that my deeds have caused you sadness,
and I have failed you.
O God, you are my God and I seek you,
my soul thirsts for you, as in a dry and weary land.

May God refresh you,
for you seek him.
May God restore you,
for your soul cries out to him.
May God revive you
with springs of living water in a dry and weary land.

Blessing and Dismissal

May God, whose ways are as high
as the heavens are above the earth,
guard us, keep us,
and bless us with his presence
as we seek to follow his ways.

Go, in the strength and power of God
to live to his praise and glory.
Now and for ever,
Amen.

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Thoughts on Isa 55 and Numbers (the digits, not the Bible book)

“For as the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts.”

Stay with me on this one. It’s relevant, I promise.

For my day job, I am a treasure-hunter. I am a solver of problems, mysteries, and enigmas, an unraveller of riddles, a demystifier of thousand-year-old conundrums, an untangler of Gordian knots. I find the ‘X’ that always, always, marks the spot.

OK, I’m a maths teacher. I was hoping I might sound like Indiana Jones.

In maths, we deal with numbers. (Well, at least initially. Some of the maths on my wall bears a striking resemblance to my Greek New Testament!)

Numbers are nice. Numbers are easy. We do numbers with tiny tots. Let’s count socks!
“One, two – two socks!” “Yay!”
“One , two, three, four – four legs on Mr Sheepy!”

It takes a while to learn that this string of random sounds is supposed to measure the ‘how many’ of something, but once we’ve got it, counting is very useful. For little kids, counting goes up to five or maybe ten, but we soon expand our ideas and find there are numbers beyond that, numbers far bigger than we could ever count. Numbers that go on and on for ever.

“My hamster’s better than your hamster.” “No he’s not. My hamster is a hundred times better than yours.” “Mine’s a thousand times better.” “Mine’s a million times better.” “Mine’s a million, billion, trillion, gazillion times better.” “Mine’s that PLUS ONE!”

And so we have the counting numbers (‘natural’ numbers, or positive integers if you want to use their posh names). They go up as far as you like, and then plus one bigger than that. And they serve very well for day-to-day purposes. “Job done?” you ask.

Not quite.

Imagine you are a first-century Roman toga seller. You have VII togas in your shop. Then someone comes in and buys VII togas. How many togas do you have left?

Ummn. Your Roman accountancy software can’t cope with that. Cumbersome numeral system and no symbol for none.

At this point I am grateful for the assistance of a Sesame Street character called The Count. He counts things, and I remember a helpful episode where he diligently counts the things that are not there. Zero. (Click to watch)

“OK”, you say, “we’ll add zero to mark the number of things I don’t have. Zero elephants on my lawn. Zero scuba-diving chipmunks in my coffee. Are we done?”

Oh no, my friend. We’ve barely even started.

Think back to when you were older than 5, but not as old as 6. How old were you? Yes, the precious halves. Don’t try to tell a five-and-a-half-year-old that they’re only five. You will get an indignant stamp of the foot and a baleful stare. “I’m five AND A HALF!” Half a year is ten percent of their life at that age. It’s worth marking. (At my age, less so.) Quarters too.

And so we come to the world of fractions. Sharing out pizzas. Chopping up oranges.

It was apple pies when I was a kid. Which I never thought was a great way to do fractions because a) I don’t like apple pie and b) my dad always had a much bigger piece than everyone else, so it doesn’t illustrate fractions well at all.

“Fine. Positive whole numbers, zero, and the fractions in between. Is that everything?”

Yeeeeee, no.

Next, we come to measuring things and we find that centimetres have lots of bits in between. Not halves or quarters, but tenths. And between those, hundredths. And between those, thousandths. We need decimals!

“I don’t mind those. Decimals are just fractions with 10 or 100 or 1000 on the bottom. Easy-peasy.”

Oh, I nearly forgot (no, I didn’t). We need the numbers for when you have fewer than zero elephants on your lawn. When there should be an elephant there and you’ve carelessly lost it. The person you borrowed it from is collecting it soon and you need one more elephant just to get up to zero. The minus numbers.

We all know about negatives. Less than zero in your bank account. Below sea level. And there are negative decimals too. The Dead Sea has an elevation of negative 430.5m.

“There was no need to bring up the negative bank account. Surely we have enough numbers now: Zero. Positive and negative whole numbers. All the fractions between those. That’s it, right?”

(Purposefully not answering the question.) These are called rational numbers. The name is supposedly because they can be written as a ratio (or fraction) of two whole numbers: this / that.

“Like 1.2 can be written as 12 / 10?”

Yes, that’s right. But I like to think they’re called rational numbers because they make sense. The rational numbers comprise all the numbers that most folks will ever need for their entire lives.

“Good. Is that it, then? Finally?”

Ummmn …. back to the apple pie again. Or perhaps I should say ‘pi’. Yes, it’s our old friend, 3-and-a-bit.

“Yeah. 3.14-ish isn’t it?”
(Bonus fact, this item was posted on 14th March, 3/14 in US format – Pi Day!)

That’s right, but it’s the ish that’s the problem. It’s weird. The kind of weird that is impossible to write down. It never ends, unlike the decimal fractions (0.375 = 375/1000). It doesn’t even repeat, like the recurring fractions (1/3 = 0.333333333333 etc.) It just carries on, forever. You could write out a million decimal places and there would still be as many left to write as there were when you started.

“That doesn’t make sense! A number can’t do that. It’s irrational!”

Yes, my friend. That’s exactly what it is. Pi is a number that cannot be written as this / that, so it’s irrational. Other irrationals include the square root of two (the number that, when you times it by itself you get two as the answer). And the square root of three, and the square root of five, six, seven, eight, ten …

“OK, OK. Enough. Positives and negatives, whole numbers, the fractions and decimals between, and the weirdo numbers between those. And zero. Please tell me we’ve finished!”

Nearly. We have now assembled the real numbers.

“Ha! I suppose you’re going to tell me there are some unreal numbers as well.”

Ummn, yeah. Kinda.

“Seriously?”

It started as a game, actually. Some mathematician was bored one day (“Really? You amaze me.”) and wondered what would happen if you could have the square root of a negative number. Now, negative numbers don’t have square roots because there is no number that, times itself, gets you a negative.

“Yes, I remember this. Negative three times negative three equals positive nine. There’s no way to get negative nine.”

But what if there were? What if we could just … imagine … some numbers that would square to give negatives. Of course, it was all just daydreaming, no useful application. I mean, these numbers weren’t even real. They were like unicorns – a pretty idea but of no practical use.

“Like a fine arts degree, you mean?”

Ooh, careful. Anyway, stop interrupting. As I was saying, imaginary numbers were an interesting idea, but there was no application in the real world.

Until there was.

From the electronic device you are reading this on, to the engineering that stops the wings falling off planes, imaginary numbers are used everywhere.

No-one was more surprised than mathematicians when these toys of recreational maths (stop looking at me like that, it’s a thing) turned out to be exactly what physicists needed to making things work.

And now we’re done. (“Really? Do you mean it? I was beginning to develop Stockholm Syndrome.”) Imaginary numbers and real numbers. You can combine them into the complex plane if you want to (“Thank you, no.”), but that’s as far as they go. To the best of my knowledge.

The interesting question, of course, is what is there beyond my knowledge?

When I was five, counting numbers went up to around twenty. I didn’t know that there was more to know.

When I was ten, numbers had decimals and fractions, and went below zero. I didn’t know that there was more to know.

When I was twenty, numbers were imaginary, complex, irrational and so on. There might be more that I don’t know about. I don’t know.

And that’s just numbers. Simple, easy numbers that we teach to tiny tots.

How much more is there to know of God?

“For as the heavens are higher than the earth,
so are my ways higher than your ways
and my thoughts than your thoughts.”

Isa 55 :1-9