Q) I am a primary PGCE student and was taught at GCSE that square numbers make squares, and triangle numbers make triangles. This seemed to make sense, until I saw a diagram of a triangle demonstrating square numbers. Can you explain?
A) The idea is there, but some of the definition is missing. Square numbers and triangle numbers are among sequences that are described as "figurative" or "polygonal" numbers; that is, each term of the sequence makes a given shape when dots are used.
So, in the case of triangle numbers, an array of dots makes an equilateral triangle and dots can be similarly arranged for a square. Of course, this sequence carries on beyond the numbers I have included diagrammatically.
Consecutive members of the triangle sequence, when added together, make a square number, so there is a relationship between them, eg 6 + 10 = 16.
The following diagram shows how triangles can demonstrate square numbers.
This happens when we look at arrangements of equilateral triangles that contain smaller equilateral triangles.
Q) What is are number bases? I was helping in an ICT lesson when the teacher said a base 2 number system was used for computers. Can you explain?
A) The number system that is base 2 is more usually called the binary system.
The number system is so called because there are only two digits: 1 and 0, whereas in our denary (decimal) system we have 10 digits.
The fact that there are only two digits makes it ideal for programming electric circuits, such as computers or traffic light systems: 0 = "off" and 1 = "on" in the circuit.
Teaching different number bases used to be a part of the old CSE and O-level exams. Though we may never use the different systems, understanding how they work leads to an understanding of our own denary system. It's a great way to practise number skills as well.
In Gulliver's Travels, Gulliver visits Houyhnhnms, a country where horses are the intelligentsia. The horses only had one "finger" on each hand - and as they had only two hands, perhaps they used a binary number system for counting.
I'm also going to suppose that they used pebbles to help them (and you) learn the place value, shown in the diagram below with two hands on a pebble for each "lump of two" that they count. They would also need some notation to mark units, so I have used another units pebble.
The Houyhnhnms place a piece of straw to identify whether the column of pebbles above it is used in the number: straw present, use that column; straw not present, don't use that column. As you can see, in this system the place value is determined by powers of two. In denary system we work with powers of 10.
Let me explain how the binary system works using this diagram. The number 1010 could be from any number system; in denary this would be one thousand and ten. The notation for recognising that this is in say, binary, is to have subscript at the end of the number, so 10102. This tells me that this number is from the base 2 number system.
Of course, when we write numbers down we don't put in all the place-value markers at the top of the column, we just write it down as we learn a feel for that number.
So probably all that the horses will have done is lay down their pieces of straw (I wonder what they will have used for zero?). You can see from the diagram that to change 10102 to our decimal system that we have (1 x 8) + (0 x 4) + (1 x 2) + (0 x 11010 so 10102 is 10 in the denary system.
This 1010 also can be interpreted in other number bases.