At one time we needed brains for long multiplication and long division, drummed into us at primary school from time immemorial. It is so long since I tried I’m not sure I can remember. Let’s try on the back of a proverbial envelope.
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| Long multiplication and long division with numbers and with pre-decimal currency |
To do it you had to be able to add up, ‘take away’ and know your times tables – eight eights are sixty four, and so on – but just about everyone born before 1980 could do these things without having to think.
Those of us still older, born before say 1960, could multiply and divide pre-decimal currency – remember, twelve pence to the shilling, twenty shillings to the pound. You had to have grown up with this arcane system to understand it. Perhaps we should have kept it. It might have put foreigners off from wanting to come here and there would have been no need for Brexit. As the example reveals, even I struggle with the division.
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| Logarithms and Antilogarithms |
Then, there were logarithms and antilogarithms, as thrown at us in secondary school. To multiply or divide two numbers, you looked up their logs in a little book, added them to multiply, or subtracted to divide, and then converted the result back into the answer by looking it up in a table of antilogs. For example, using my dinky little Science Data Book, bought for 12p in 1973:
To multiply 2468 x 3579:
log 2468 = 3.3923; log 3579 = 3.5538; sum = 6.9461; antilog = 8,833,000
To divide 3579 by 24:
log 3579 = 3.5538; log 24 = 1.3802; subtraction = 2.1736; antilog 149.1
It’s absolute magic, although the real magicians were individuals like Napier and Briggs who invented it. How ever did they come up with the idea? It was not perfect. Log tables gave only approximate rounded answers and it was tricky handling numbers with different magnitudes of ten (represented by the 3., 6., 1. and 2. to the left of the decimal points), but it was very satisfying. You needed ‘A’ Level Maths to understand how they actually worked, but not to be able to use them. Some also learned to use a slide rule for these kinds of calculations – a mechanical version of logarithms – but as I never had to, I’ll skip that one.
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| A Slide Rule |
Due to a hopeless lack of imagination, I left school to work for a firm of accountants in Leeds. Contrary to what you might think, our arithmetical skills were rarely stretched beyond adding up long columns of numbers. We whizzed through the totals in cash books and ledgers, and joked about adding up the telephone directory for practice. The silence of the office would be punctuated by cries of torment and elation: “oh pillocks!” as one desolate soul failed to match the totals they had produced moments earlier, or a tuneless outbreak of the 1812 Overture as another triumphantly agreed a ‘trial balance’ after four or five attempts.
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| A 1960s Sumlock Comptometer. |
But when it came to checking pages and pages of additions we had comptometer operators. Thousands of glamorous girls left school to train as Sumlock ‘comps’, learning how to twist and contort their fingers into impossible shapes and thump, thump, thump through thousands of additions in next to no time without ever looking at their machines. By using as many fingers as it took, they could enter all the digits of a number in a single press. It probably damaged their hands for life. I still don’t understand how they did it. There was both mystery and glamour in going out on audit with a comp.
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| A 1950s Friden Electromechanical Calculator |
Back at the office we had an old Friden electro-mechanical calculating machine. What a beast that was. I never once saw it used for work, but we discovered that if you switched it on and pressed a particular key it would start counting rapidly upwards on its twenty-digit register.
“What if we left it on over the bank holiday weekend?” someone wondered one Friday. “What would it get to by Tuesday?”
Fortunately we didn’t try. It would probably have burst into flames and set fire to all the papers in the filing room. But we worked it out (sadly not with the Friden). It operated at eight cycles per second. So after one minute it would have counted to 480, after one hour to 28,800, and after one day to 691,200. So if we had started it at five o’clock on Friday, it would have got to 2,534,400 by nine o’clock on Tuesday morning. So, counting at eight per second gets you to just two and half million after three and a half days! It shows how big two and a half million actually is.
The obvious questions to us awstruck nerdy accountant types were then “what would it get to in a year?”– about two hundred and fifty million, and “how long would it take to fill all twenty numbers in the top register with nines?”– about thirty nine million million years. As the building was demolished in the nineteen eighties it would have been switched off long before then. But what would it have got to?
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| An ANITA 1011 LS1 Desktop Calculator (c1971) |
The first fully electronic machine I saw was a late nineteen-sixties ANITA (“A New Inspiration To Accounting”, one of the first of many truly cringeworthy acronyms of the digital revolution) which looked basically like a comptometer with light tube numbers. Then, fairly quickly with advances in integrated circuits and chip technology, came the ANITA desk top calculator followed by pocket handhelds that could read HELLHOLE, GOB and BOOBIES upside down, and 7175 the right way up. Intelligence was as redundant as comptometer operators. We revelled so much in our mindless machine skills that I once saw a garage mechanic work out the then 10% VAT on my bill with a calculator, and get it wrong and undercharge me. It can still be quicker to do things mentally rather than use a calculator.
Around 1972, my dad saw one of the first pocket calculators for sale in Boots. It could add, subtract, multiply and divide, pretty much state of the art for the time, but at £32 (about £350 in today’s money) and not as compact as now, it required large pockets in more ways than one. I told him it was ridiculously overpriced. Infuriatingly, he ignored me and bought one. On the following Monday they reduced the price down to just £6. It was his turn to be annoyed but the store manager refused to give a refund. He stuck with that calculator for the next thirty years.
How often now do we even use calculators? Not a lot for basic arithmetic. Do we ever doubt the calculations on our computer generated energy bills and bank statements? Do we check the VAT on our online purchases? Do accountants ever question the sums on their Excel spreadsheets? Just think, a fraction of a penny here, another there, carefuly concealed, embezzlement by a million roundings, it could all add up to a nice little earner.
I believe the above images to be in the public domain except for the first which is mine.
