As readers of this blog will know, I enjoy trying to find ways of getting our heads around large numbers and huge quantities. In my experience children in primary schools also find large numbers fascinating, and, with the help of a calculator and some approximate mental calculations, we can play with big numbers and do some interesting mathematics.
This last week saw the landing of 'Philae' on Comet 67P/Churyumov–Gerasimenko. The newspaper reports told us that this comet was 300 million miles away. That's a long way, I'm sure, but how can we understand a distance like this? I always try to connect these big numbers with our own experience.
So, let's imagine I start driving my car at an average speed of 45 mile per hour. How long is it going to take me to clock up 300 million miles, assuming I have a co-driver and we manage to drive night and day non-stop?
Well, let's see 45 mph is 45 × 24 miles per day, which is 1080 miles per day.
That's about 1080 × 365.25 miles per year, which looks like approximately 400,000 miles per year.
So to find how many years it will take me ... careful with all these zeros ... we need 300,000,000 divided by 400,000, which is 750 years!
So, put it like this: imagine I had started this epic drive in 1264, back in the Middle Ages, during the reign of Henry III, in the middle of the Crusades, when Marco Polo, that intrepid traveller, was 10 years old; I would have continued on driving through all the subsequent Henrys, through the reigns of the Tudors and Stewarts, and the Civil War; I would have still been driving when Beethoven was born and when he died; and when Queen Victoria came to the throne and when she died; and still driving during the two World wars, mysteriously passing the day of my own birth, and right through to the present day, and I would be now just closing in on the 300 million miles target!!
Yes, it's a long way to Comet 67P/Churyumov–Gerasimenko!
Showing posts with label large numbers. Show all posts
Showing posts with label large numbers. Show all posts
Sunday, 16 November 2014
Tuesday, 24 December 2013
How large is the largest known prime number?
In editing Mathematics Explained for Primary Teachers I had to update the bit about the largest known prime number, because clever people with even cleverer computers keep discovering larger ones. So far every new edition of the book has required an update!
At the time of writing, to my knowledge, the largest known prime number was ‘257,885,161 − 1’. This means 57,885,161 twos multiplied together, minus 1: which produces a number with over 17 million digits.
How large is this? It's very difficult to conceive of a number with 17 million digits. So, I came up with the following idea. Imagine trying to write it down. The fifth edition of my book will contain over 400 pages. Just to print all 17 million-plus digits in the largest known prime number would require ten books of that size!
At the time of writing, to my knowledge, the largest known prime number was ‘257,885,161 − 1’. This means 57,885,161 twos multiplied together, minus 1: which produces a number with over 17 million digits.
How large is this? It's very difficult to conceive of a number with 17 million digits. So, I came up with the following idea. Imagine trying to write it down. The fifth edition of my book will contain over 400 pages. Just to print all 17 million-plus digits in the largest known prime number would require ten books of that size!
Monday, 23 September 2013
Grains of rice on chessboard problem
My grandson, Jack, now 12, called this week and told me about something that he had learnt at school: it was the old problem of grains of rice on a chessboard. I was pleased to learn that it was his history teacher, Mr Pearce, not his maths teacher, who had told him about this nice piece of mathematics and caught his imagination!
The story is about someone requesting 1 grain of rice on the first square of a chessboard, 2 on the next, 4 on the next, 8 on the next, and so on, doubling the number each time. The total number of grains of rice involved turns out to be an astounding number; much, much more than the number of grains of rice produced in the whole world in one year.
The mathematical series involved here is: 1 + 2¹ + 2² + 2³ + 2⁴ + ... + 2⁶³. The total of this is actually just 1 less than 2⁶⁴.
At a quick estimate, I made it about 1.8 × 10¹⁹. That's 18 followed by 18 zeros. Jack (and his Dad) seemed a bit disappointed that the number wasn't bigger than this!
How might we get a sense of the size of a number like this? I suggested we think about how long it would take to count this number of grains of rice, counting at the rate of 2 per second. A quick calculation on a calculator gave us the result: counting day and night, never stopping, it would take a person 300,000,000,000 years! Three hundred, thousand, million years! OK? It's a big number, yes?
The story is about someone requesting 1 grain of rice on the first square of a chessboard, 2 on the next, 4 on the next, 8 on the next, and so on, doubling the number each time. The total number of grains of rice involved turns out to be an astounding number; much, much more than the number of grains of rice produced in the whole world in one year.
The mathematical series involved here is: 1 + 2¹ + 2² + 2³ + 2⁴ + ... + 2⁶³. The total of this is actually just 1 less than 2⁶⁴.
At a quick estimate, I made it about 1.8 × 10¹⁹. That's 18 followed by 18 zeros. Jack (and his Dad) seemed a bit disappointed that the number wasn't bigger than this!
How might we get a sense of the size of a number like this? I suggested we think about how long it would take to count this number of grains of rice, counting at the rate of 2 per second. A quick calculation on a calculator gave us the result: counting day and night, never stopping, it would take a person 300,000,000,000 years! Three hundred, thousand, million years! OK? It's a big number, yes?
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