2+2=?
This was part of the initial draft for the page Nothing in life is simple. In the end, I decided to edit most of it out … but thought I would include as a “subpage” … just in case anyone is interested.
As children we learn the rules of basic arithmetic: 2+2=4, and so on. If we have two toys and are given another two toys, how many do we have altogether? Answer: four toys. This means that 4-2=2. If we have four toys and give away two toys, how many are left? Answer: two toys.
Simple, isn’t it?
That’s fine, but what happens if we get a question like 2-4? We learn that you can’t do it. If you have 2 toys, then you can’t give four away. It’s impossible.
But then we change the situation. If the temperature yesterday was 2° and today it is 2 degrees warmer, what is the temperature today? 2+2 still equals 4. If the forecast for tomorrow says that it will be 2 degrees colder than today, what will the temperature be tomorrow? That’s easy, 4-2 = 2, right? If the following day it the temperature falls by four degrees, what will the temperature be then? So the question is 2-4 – can’t do it – right? Wrong! This is where we learn about “negative numbers” and that 2-4 = -2
The question is the same in arithmetical terms, but the contexts are different so the rules are changed and the answers are different. In one case it’s “impossible” – you “can’t do it” – and in the other case the answer is -2.
OK – it’s a bit more complicated – but it’s still quite simple, isn’t it?
(As an aside, notice that 3+2 gives the same answer as 2+3 … 5, the order of the numbers doesn’t matter when we are adding numbers … but it does when we are subtracting. The answer to 5-3, (that is, 2),is not the same as the answer to 3-5, (that is, -2))
We also learn our “times table”. If 2+2=4, that means that two lots of 2 makes four, or 2×2=4. If we by packs of sweets containing 2 Mars bars, then how many mars bars do we actually have. There are 2 packs, and each pack has 2 Mars bars, so the answer is: 2×2=2.
Just as addition and subtraction are opposites, so are multiplication and division. If 2×2=4, then 4¸2=2. So, if we have 4 Mars bars and share them between us, how many do we each get? Answer: 4¸2=2 – 2 Mars bars each.
Likewise, if 2+2+2=6, that means that 3 lots of 2, or 3×2, also equals 6 and 6¸3=2 and 6¸2=3. If I buy 3 packs Mars bars with 2 bars in each, I will have 6 Mars bars, (3×2), and if we share them between us, (6¸3) then we will get 3 each.
So far, so good? Simple?
OK, there’s a bit of a complication, in that we also sometimes use the symbol / instead of ¸ so can we have 6/3=2 and 6/2=3. This is useful for when we come to deal with fractions – “vulgar” fractions … but let’s not go into that right now. (As an aside, I have often wondered what “polite” fractions were … but that’s a linguistic complication rather than a mathematical one.)
(As an aside, notice that 3×2 gives the same answer as 2×3 … 6, the order of the numbers doesn’t matter when we are multiplying numbers … but it does when we are dividing. The answer to 6/3, (that is, 2),is not the same as the answer to 3/6, (that is, ½) – but that’s another story.)
What about those funny “negative numbers”? How do they behave when it comes to multiplication and division? As it happens, it’s not too difficult. Consider 3x(-2) … this means 3 lots of (-2), or -2 -2 -2 which gives us -6. It doesn’t take long before we start learning parrot fashion such rules as “plus times a plus equals plus, plus times a minus equals minus, minus times a minus equals plus”.
A little later on we learn about square numbers – the result when we multiply a number by itself – such as 22 and 3×3. This is called “squaring. So, 2×2 is called 2-squared and written as 22. That means that 22.=4 Likewise 3×3 is called 3-squared and written as 32 and equals 9.
Now, if addition and multiplication have opposites, then does “squared” have an opposite? Yes, of course it does – it’s called a “square root”. – and we use a special symbol, √, to show that we want to find a number, which when multiplied by itself gives us a particular answer. If 22=2×2=4, then √4=2. Likewise, as 32=3×3, which means that √9=3.
Hey, this is all simple stuff, isn’t it?
OK – I know that lots of people don’t like Maths … and it starts going to get worse because this is where the complications really start mounting up …
One of the problems with square numbers is that only a few of them are a few of them are nice and neat. For example … 22=4, 32=9, 42=16 … and so on. There’s a gap between each of them – and as the numbers get bigger, so the gap gets larger. So, to put it another way … √4=2, √9=3, √16=4 and so on, but what about the numbers in between?. What about √5, √6, √7 and √8 for example. Well there are square roots for these numbers, but they are what is called “irrational numbers”, (I said it got complicated), which means that they can’t be written exactly except with that funny “surd” symbol – √.
Now, what about those funny “negative numbers”? How do they behave when it comes to squares and square-roots?
Well, when it comes to squares – it’s simple, (isn’t it?) (-2)2 means -2 x -2 which equals 4 … remember, “minus time a minus equals plus”. So, there are two possible answers to what number when multiplied by itself gives an answer of 4? – 2 and -2. The square root of 4, (√4), can be either 2 or -2 and we learn to right this as √4=±2.
But what about √-4 ? What number when multiplied by itself gives and answer of -4? Now, because of the rule that “plus times a plus equals plus, plus times a minus equals minus, minus times a minus equals plus” then any number which is multiplied by itself is bound to give a positive answer. As a result, we learn that, (just like 3-5), it can’t be done.
It’s only later that we learn that it is indeed possible, when we learn about “complex numbers” and the “imaginary number”, i, (i=√-1).
And we haven’t started to consider the question of 2-2, 3-2 … and so on.
By the way, when I said that 2+2=4, then I could well have added that it could in fact be anywhere between 3 and 5!
Really !!!???!!!!
It’s all a question of accuracy, what the mathematicians call “sig figs.”, or significant figures.
When dealing with discrete objects, like apples or toys, then 2 means two and 2=2 does equal 4. When, on the other hand, we are dealing with “continuous data” such as lengths in centimeters – and even with money – then we often round off. The usual method of rounding is to round anything like 1.5, 1.6, 1.7, 1.8 and 1.9 up to 2 and anything like 2.1, 2.2, 2.3 and 2.4 – up to 2.4999999999…. down to 2. so 2+2 could mean anything from 1.5+1.5 to 2.4999999999…+2.4999999999… hence the answers of 3 and 5.
To make matters worse, there are other ways of rounding off … not “to the nearest” but even “residuals” get rounded up and odd “residuals” are rounded down. So, 1.2, 1.4, 1.6, 1.8, 2, 2.1, 2.3, 2.5, 2.7 and 2.9 are all rounded off to 2. In other words, 2+2 could mean anything between 1.2+1.2 and 2.9+2.9 – giving answers between 2.4 and 5.8.
I hope that I have made the point. What looks like something simple – may not be quite as simple as it appears. A lot depends on the context … in a different setting, different rules may apply … and lead to different conclusions.
Bishkek