Multiplication and Division Symbols, Simultaneous Equations, Bootswatch

1. Multiplication and Division Symbols

It was an English mathematician, William Oughtred, in 1631 who used × to represent multiplication and : to represent division. See Enlightening Symbols by Joseph Mazur, p83.

I find it interesting that : is only used in the UK for ratios, rather than division itself whereas in Finland it is still used to indicate division.

2. The Flower of Thymaridas

A rule to solve some sets of simultaneous equations was known in early Pythagorean times. It was known as the Flower of Thymaridas. For example for four variables, you will need to know the sum of all four variables and then to find one particular variable, you will need to know the sum of that variable and each of the other three. For example, the sum of all four is 20, and the sum of the quantity we want to know and the second quantity is 7, and the sum with the next quantity is 6 and the sum with the final quantity is 15. In algebraic terms, if x is the quantity we want to know, we have the following.

     x + y + z + w = 20  (1)
     x + y         =  7  (2)
     x     + z     =  6  (3)
     x         + w = 15  (4)

To find the desired quantity, sum the sums of the pairs, and then subtract the sum of the four numbers. We then divide by the number of equations we added minus one, so in our example, we sum 7, 6 and 15 to get 28 and then subtract 20 leaving 8. This is then divide by 3-1 = 2 so our unknown quantity is 8/2 = 4. In algebraic terms, this means the following.

(2) + (3) + (4)    3x + y + z + w = 28  (5)
                    x + y + z + w = 10  (1)
                   ___________________
(5) - (1)          2x             =  8  (6)
(6)/2               x             =  4  

See Enlightening Symbols by Joseph Mazur, p93. and Wikipedia

3. Bootswatch

Instead of using Bootstrap straight out of the box or spending a considerable amount of time customising it yourself, you can use themes created by other people such as at Bootswatch.