It is some time since I last wrote about maths and the last blog entry happened to contain some support for maths as an academic subject free from any practical use. However there are so many practical uses. Someone told me that they did not see a practical use for algebra. I would say that clear thinking is one response but I suppose you could say that about any subject that has no obvious practical connection.
You need to get the right change. You need to know that the amount you are being charged is the right amount. You need to know the odds when you place a bet. It may be that we generally get the right change. It may be that we can trust people to charge us the right amount and it may be that we can place a bet for the fun of it with the full knowledge that the bookmaker should win. But with maths we do all these things with our eyes open.
When Noel Edmonds asks if there is a deal the contestant asks for advice from the other contestants. I admit that the judgement is not purely a mathematical one, but so little of that advice relates to maths and it should.
We know that the bookmaker has stacked the odds in their favour, but with mathematical support we can enjoy a bet (or not) knowing the odds. I'd bet that you don't find many with maths degrees at gamblers anonymous.
That sums it up
Tuesday 20 March 2012
Thursday 7 May 2009
Simultaneous Equations
I like to think of maths as a means to practice clear thinking. There are those who criticise pure maths because it is not practical. People leave school and there are many mathematical processes that will never be used again. However there are lots of maths that is needed in day to day life. We need to get the right change, we need to buy the right amount of timber or sweets or dog food. We need to know how many minutes are in an hour, and we need to know about numbers when we drive at 30mph.
Maths is all around us but it is also a clear way of thinking. I have written about clear thought when talking about deal or no deal, and went on to talk about a system that is a little more complex - the Monty Hall problem. Well what if you sent two people to the chip shop and the first person bought three fish and two lots of chips and it costs £7.40 The second person two fish and one lot of chips and it costs £4.60 Now a third person wants fish and chips. How much is it going to cost them? If you don't know about simultaneous equations you have to phone the shop. If you do know about them the answer is in your grasp.
Let's put it mathematically and fish becomes f and chips becomes c. You end up with two equations: 3f + 2c = 7.4 and 2f + c = 4.6 The thing to do is learn to manipulate two sets of equations, and the question that you have to ask is what is the simplest way to end up with just fs or just cs? You can't simply add the equations together of take one from the other but if you do take the second from the first you get a third equation f + c = 2.8 Now take this from the second equation and you get f = 1.8 and you can now put this figure in any of the equations and you know the price of chips.
The actual costs of fish and chips doesn't matter. What does matter is that you know how to manipulate simultaneous equations so that you get your answer.
That sums it up
Maths is all around us but it is also a clear way of thinking. I have written about clear thought when talking about deal or no deal, and went on to talk about a system that is a little more complex - the Monty Hall problem. Well what if you sent two people to the chip shop and the first person bought three fish and two lots of chips and it costs £7.40 The second person two fish and one lot of chips and it costs £4.60 Now a third person wants fish and chips. How much is it going to cost them? If you don't know about simultaneous equations you have to phone the shop. If you do know about them the answer is in your grasp.
Let's put it mathematically and fish becomes f and chips becomes c. You end up with two equations: 3f + 2c = 7.4 and 2f + c = 4.6 The thing to do is learn to manipulate two sets of equations, and the question that you have to ask is what is the simplest way to end up with just fs or just cs? You can't simply add the equations together of take one from the other but if you do take the second from the first you get a third equation f + c = 2.8 Now take this from the second equation and you get f = 1.8 and you can now put this figure in any of the equations and you know the price of chips.
The actual costs of fish and chips doesn't matter. What does matter is that you know how to manipulate simultaneous equations so that you get your answer.
That sums it up
Tuesday 5 May 2009
Factorising 2
After the last blog there should now be now no fear of factorising. You may wonder, though, how far you can go with it. What are the smallest numbers that you can factorise a large number into? The answer is, the prime numbers. If you find any factor that isn't prime, that factor can in turn be factorised into prime numbers. Quite often you will be asked to find all the prime factors of a number.
For 28, for instance, 14 and 7 and 4 and 2 are all factors, but the 14 and the 4 are not prime. The prime factors are 7 and two 2's. Note that a single 2 is not enough, because 7 x 2 is just 14, only half of what we need. 28 has three prime factors, and it just happens that two of them are the same as each other. 28 = 7 x 2 x 2.
If I give you the question factorise 4n + 8, you need to take out the common factor. There is a common factor of 4 so factorising 4n + 8 is simple. It is 4(n + 2). I hope you followed that. I had to put a letter in the equation because that is what algebra is all about.
As long as there's just one letter, and it isn't squared or anything tricky like that, factorising is just a matter of finding the common factors, the ones that divide into both the constant part (8 in the example above) and the part which involves a variable represented by a letter (4n in the example). It can get trickier if there are squares or other powers, or if there is more than one variable, but that's a matter for another blog. Even in those cases the principle is still the same - you're trying to find simple expressions or numbers which can divide into the original expression.
That sums it up
For 28, for instance, 14 and 7 and 4 and 2 are all factors, but the 14 and the 4 are not prime. The prime factors are 7 and two 2's. Note that a single 2 is not enough, because 7 x 2 is just 14, only half of what we need. 28 has three prime factors, and it just happens that two of them are the same as each other. 28 = 7 x 2 x 2.
If I give you the question factorise 4n + 8, you need to take out the common factor. There is a common factor of 4 so factorising 4n + 8 is simple. It is 4(n + 2). I hope you followed that. I had to put a letter in the equation because that is what algebra is all about.
As long as there's just one letter, and it isn't squared or anything tricky like that, factorising is just a matter of finding the common factors, the ones that divide into both the constant part (8 in the example above) and the part which involves a variable represented by a letter (4n in the example). It can get trickier if there are squares or other powers, or if there is more than one variable, but that's a matter for another blog. Even in those cases the principle is still the same - you're trying to find simple expressions or numbers which can divide into the original expression.
That sums it up
Monday 4 May 2009
Factorising
If you are factorising a number then you are looking for the factors, where factors are the numbers which when multiplied together give you the number you started with. Think of a number, any number. OK I have thought of one. It is 27. There are always four factors to any number 1 and that number, so for 27, 1 and 27 are factors. The other two factors are the equivalent numbers but this time with a negative sign, because a negative times a negative is a positive. So the two other factors of 27 are -1 and -27.
If I had chosen an even number there would have been other factors because 2 would have been there along with half the original number (and then you would have the same numbers but with negative signs). In the case of 27 there are also the factors 3 and 9. You can find this by trial and error. Obviously miss out all the even numbers. 5, 7, 11, 13 are not factors. It didn't take long to work that out so I now know that the only factors of 27 are 1,3, 9 and 27.
That sums it up
If I had chosen an even number there would have been other factors because 2 would have been there along with half the original number (and then you would have the same numbers but with negative signs). In the case of 27 there are also the factors 3 and 9. You can find this by trial and error. Obviously miss out all the even numbers. 5, 7, 11, 13 are not factors. It didn't take long to work that out so I now know that the only factors of 27 are 1,3, 9 and 27.
That sums it up
Sunday 3 May 2009
Algebra
Any subject can be fantastic if you know how to do it and a teacher marks your work and then tells you how well you have done. The beauty of mathematics is that if you tell your teacher that 2 + 2 = 4 then you are right and nobody can take that away from you. I was watching Britain's got talent and you may be the person who gets the audience on its feet, but there may be some people who don't like what you do and turn the TV off. Just look at this algebraic equation and see how simple it is.
Algebra is a branch of mathematics that substitutes letters for numbers. You can have some equations like x + y = 3 and x - y = 1, and you may be able to see at once that x is 2 and y is 1. Very often in maths it is not the answer that is important but how you get the answer. If you know how to do something then you can always do it but you can always make simple errors however good your maths is.
When you see an equals sign you know that the left side of the equation is equal to the right. If you add something to the left then it remains equal if you add the same number to the right. If you now add those two equations together you get 2x (y-y is zero) = 3 +1. If that is right then you divide both sides by 2 and you get x = 2. I hope you followed each step.
That sums it up.
Algebra is a branch of mathematics that substitutes letters for numbers. You can have some equations like x + y = 3 and x - y = 1, and you may be able to see at once that x is 2 and y is 1. Very often in maths it is not the answer that is important but how you get the answer. If you know how to do something then you can always do it but you can always make simple errors however good your maths is.
When you see an equals sign you know that the left side of the equation is equal to the right. If you add something to the left then it remains equal if you add the same number to the right. If you now add those two equations together you get 2x (y-y is zero) = 3 +1. If that is right then you divide both sides by 2 and you get x = 2. I hope you followed each step.
That sums it up.
Saturday 2 May 2009
Square Roots
What is the positive square root of 9? This was a question today (1st May) on the Weakest Link. So learn your maths and who knows, you could win a cash prize on TV. I hope that you know the answer to this question. It is three, but why did Ann Robinson ask for the positive square root? She asked because a positve number times a positive number is a positive number. This means that three time three is nine. There is another answer. A negative number times a negative number is a positive number. So if the question is 'what is the square root of 9' there are two answers, three and negative three.
Do you know the square root of zero? The answer is zero. If you multiply anything by zero you get zero, and this includes zero. we know we can get square roots of positive numbers and zero, but can we have a square root of a negative number? If we start with minus nine and look for the square root, we can't have three because 3 x 3 = 9. We can't have minus three because -3 x -3 = 9. I hope that everything in this blog has been simple for you because when you deal with square roots of negative numbers you have to use something called complex numbers, but that is the subject for another blog.
That sums it up
Do you know the square root of zero? The answer is zero. If you multiply anything by zero you get zero, and this includes zero. we know we can get square roots of positive numbers and zero, but can we have a square root of a negative number? If we start with minus nine and look for the square root, we can't have three because 3 x 3 = 9. We can't have minus three because -3 x -3 = 9. I hope that everything in this blog has been simple for you because when you deal with square roots of negative numbers you have to use something called complex numbers, but that is the subject for another blog.
That sums it up
Friday 1 May 2009
Sequences
A sequence is an ordered list of numbers. It could be made up of integers but it doesn't have to be. I am looking at a GCSE question that gives you a sequence composed of 5 9 13 17 and 21. You are asked to find an expression for the nth term in this sequence. You have to find the pattern. I will look at how to delve deeper into patterns in later blogs but this one does not need a deep explanation. How do you get from 5 to 9? You have to add 4. How do you get from 9 to 13? You add 4. Just follow the sequence and you see that you do this each time.
You have to find the nth term where n can be any number and you know that if n=1 you have 5, so an expression for the nth term has to start 5... The next thing you know is that you add 4 each time so the 2nd term is 5 + 4. Now this is the second term but there is only one lot of 4 not two. So the expression for the nth term is 5 + 4(n-1)
That sums it up
You have to find the nth term where n can be any number and you know that if n=1 you have 5, so an expression for the nth term has to start 5... The next thing you know is that you add 4 each time so the 2nd term is 5 + 4. Now this is the second term but there is only one lot of 4 not two. So the expression for the nth term is 5 + 4(n-1)
That sums it up
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