|
|
| (35 intermediate revisions by one other user not shown) |
| Line 1: |
Line 1: |
| | | #redirect [[Using TeX Notation 3]] |
| ==Brackets==
| |
| | |
| As students advance through Maths, they come into contact with brackets. Algebraic notation depends heavily on brackets. The usual keyboard values of ( and ) are useful, for example:
| |
| | |
| <math>d = 2 \ \times \ (4 \ - \ j)</math>
| |
| | |
| This is written as:
| |
| | |
| $$ d = 2 \ \times \ (4 \ - \ j) $$
| |
| | |
| Usually, these brackets are enough for most formulae but they will not be in some circumstances. Consider this:
| |
| | |
| <math>4x^3 \ + \ (x \ + \ \frac{42}{1 + x^4})</math>
| |
| | |
| Is OK, but try it this way:
| |
| | |
| <math>4x^3 \ + \ \left(x \ + \ \frac{42}{1 + x^4}\right)</math>
| |
| | |
| This can be achieved by:
| |
| | |
| $$ 4x^3 \ + \ \left(x \ + \ \frac{42}{1 + x^4}\right) $$
| |
| | |
| A simple change using the \left( and \right) symbols instead. Note the actual bracket is both named and presented.
| |
| | |
| ==Ellipsis==
| |
| | |
| The Ellipsis is a simple code:
| |
| | |
| <math>x_1, \ x_2, \ \ldots, \ x_n</math>
| |
| | |
| Written like:
| |
| | |
| $$ x_1, \ x_2, \ \ldots, \ x_n $$
| |
| | |
| A more practical application could be:
| |
| | |
| Question:
| |
| "Add together all the numbers from 1 <math>\ldots</math> 38.
| |
| What is an elegant and simple solution to this problem?
| |
| Can you create an algebraic function to explain this solution?
| |
| Will your solution work for all numbers?"
| |
| | |
| Answer:
| |
| The question uses an even number to demonstrate a mathematical process and generate an algebraic formula.
| |
| | |
| {| class = "nicetable"
| |
| |-
| |
| | Part 1:
| |
| | Part 2.
| |
| | Part 3.
| |
| |-
| |
| |
| |
| <math>1. \ 1 \ + \ 38 \ = \ 39</math>
| |
| | |
| <math>2. \ 2 \ + \ 37 \ = \ 39</math>
| |
| | |
| <math>3. \ 3 \ + \ 36 \ = \ 39</math>
| |
| | |
| <math>\ldots</math>
| |
| | |
| <math>19. 19 \ + \ 20 \ = \ 39 </math>
| |
| | |
| <math>\therefore x \ = \ 39 \ \times \ 19 </math>
| |
| | |
| <math>\therefore x \ = \ 741 </math>
| |
| | |
| | |
| |An algebraic solution might read something like:
| |
| <math>t = (1 + n) \times n/2 </math>
| |
| | |
| Where t = total and n = the last number.
| |
| | |
| |The solution is that, beginning from each end, the numbers are added and then multiplied by the number of different combinations.
| |
| The answer must depend on the number, <math>\frac{n}{2}</math> being a whole number. Therefore, the solution will not work for an odd range of numbers, only an even range.
| |
| | |
| | |
| |}
| |
