# Sum

Professor Addington posed the following problem to her student, Sammy Adder: “Can you explain why 1 + 1 = 10?”

“Addition should be a straightforward exercise, shouldn’t it? I was taught that 1 + 1 = 2 so what is Prof Addington on about?” Sammy wondered.

Without hesitation, Sam Summers interjected, leaving his classmate with a puzzled look on his face; “It’s because you’re counting in base 2, not base 10!”

“You certainly know your sums, don’t you, Sam Summers!” Prof Addington said.

It is true that 1 + 1 = 2. In fact, it’s the basis of our mathematical education. Right from the start we’re taught about numbers and how to add them up. But it is equally true that 1 + 1 = 10 or even that 1 + 4 = 10 if you’re counting in base 5.

But it is also true that blue + yellow = green, isn’t it? This is another undeniable fact and it has to do with how we interpret colours. Since it is the frequency of a light wave that determines its colour (see Spectrum), then the addition of two waves of light will result in a new wave whose frequency is the sum of the other two. And when this resulting wave interacts with the receptors in our retina, our brain interprets this as the colour ‘green’.

Yellow and blue when added together gives green.

So addition is not limited to numbers only. We can pretty much add anything we want. And the things we’re adding up do not need to be countable either. Let’s consider another example, which is not too dissimilar from the addition of colours. Sound waves can also be added together. And what they result in is what we interpret as ‘beats’. And since the ways in which we can sum the waves up is not limited, we can produce any beat we want. But of course, not all of them would necessarily be audible since the human hearing range is itself limited.

As you can imagine, addition is not a trivial trick. It’s more than a simple arithmetic tool. We can look beyond adding two numbers together and add things like waves or even shapes. By adding two right-angle triangles together you get a square. Try it! And by adding two squares together you get a rectangle. From triangles to rectangles, it’s as easy as 1, 2, 3! Now, there are other kinds of triangles that you can use in this little addition exercise. If you add six equilateral triangles together you end up with a regular hexagon. And if you were to draw those triangles on a computer and animate them and make them jiggle around or move about, what you have is essentially the basic of a computer-generated image. All that is left to do is make this process a little bit more sophisticated and add a splash of colour and, voila, you’ll dazzle everyone with your mind-blowing CGI!

Addition is a really interesting concept indeed. This little ‘+’ sign holds a lot of power. In fact, there is a whole area of study dedicated to this mathematical operator. So, we can ignore the things we are actually adding together and focus more on the process of addition itself. What exactly does this symbol ‘+’ do? That is the question.

Good mathematicians are known to be lazy – in the sense that they would find the shortest, quickest and most efficient way to reach a solution rather than toil away in search of the answer. Addition allows you to combining almost anything together. But if you had to repeat that process over and over again then you might as well use the multiplication operator. Multiplication, you see, is simply an extension of addition. And taking this one step further, integration in calculus is yet another manifestation of addition. When you are working out the area under a curve using integral calculus, what you are simply doing is adding small section of the area bounded by the curve and the axes.

Area under a curve can be approximated by adding up small rectangles.

How about subtraction, you ask? Well it is yet another form of addition where, this time, it is about adding opposites. The simplest example I can think of is this: 2 – 1 = 1. We can think of this equation as follows: 2 + (-1) = 1. We are effectively adding 2 and the opposite of 1, i.e. -1.

Don’t worry, I haven’t forgotten about division, the fourth of the most used mathematical operators. Again, it is not surprising that with the concept of addition and opposites we can come up with the concept of division. How can we get to 50 from 100? One way to do it is to divide 100 by 2, which is essentially the same as subtracting 1 from 100 fifty times. Sometimes the answer we are looking for might not be a whole number. For example, 7 divided by 2 is 3.5. Again, we can use the addition of opposites, or subtraction to reach 3.5 but this time we use fractions or parts of a whole number.

Speaking of adding up fractions, here is an interesting puzzle. Usually when you keep on adding stuff together, the total number gets bigger and bigger. 1 + 1 + 1 + … and so on will lead to a very, very large number. But sometimes, the things you are adding up are such that their sum remains a small, finite number. Take an apple, for instance. Add to it half of another apple. You end up with 1½ apple. To this 1½ add ¼ to get 1¾ apple. Next add 1/8th then 1/16th and so on. How many apples would you end up with? The answer is 2. There are other sequences of numbers (whole and fraction) that you can play with to find out what comes out of their summation.

Adding fractions continuously to obtain a whole, yet finite, number.

Two apples.

The point is, from the concept of addition we can come up with other kinds of mathematical operators such as -, × and ÷. There are several other interesting features about addition, which we can explore but I will leave that up to you to investigate – the fun in mathematics is actually in finding things out.

Mathematics is full of these seemingly trivial concepts but they are actually fundamental to so many, more complex, ideas. They are far from insignificant. There is so much to be discovered about this fascinating subject. Don’t be fooled by its complexity: at the most fundamental level you have neat little ideas, like the inconspicuous + sign, which holds a lot of power. This power is yours to behold…