The manner in which we count is based on the number of fingers (digits) that we have.

Our number system is the product of centuries of development . The symbols originated with the Hindus and the name from the Romans. Decimal, means tenth or tithe.

The numbers are put together so that the posi ...read more.

Middle

The algorithm was presented without proof to avoid others claiming the idea as their own. The method is clearly the same as the formula method of solution. He showed the solution of particular equations, for example: x2 - 10x = -9. He called the numbers by the names:

1 is "the square"

-10 is "the unknown"

-9 is "the absolute number"

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Brahmagupta's work yields only one solution for each quadratic equation. However, by the 12th century, scholars were identifying both solutions. Problems were now being written to teach mental skills rather than to solve problems from astronomy.

Another Indian scholar, Bhaskara set a problem such as this one:

The fifth part less three, squared, of a troupe of monkeys had gone to a cave and one monkey was in sight having climbed a tree. Say how many monkeys there are.

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Although the two values of the unknown were described by Bhaskara, one value was dismissed as incongruous.

If the number of monkeys is 5, the number going to the cave is negative. This is clearly not a solution to the riddle. What is the only

reasonable solution?

The monkey problem can be written as a modern equation:

which, rewritten, becomes x2 - 55x + 250 = 0

These ideas were translated into symbols in the 17th Century and should be familiar to all students of mathematics.

ax2 + bx + c = 0

x =

-b ± (b2 - 4ac)

2a

Factorisation

If the product of two numbers p and q equals zero, then either p or q (or both) are zero.

If

p × q

= 0

then

either

p

= 0

or q

= 0

orp

= q = 0

This means that if a quadratic equation can be factorised, it is possible to extract linear equations which lead to solutions. These solutions are often referred to as roots of the equation.

x - 2.3) = x2- 3.8x + 3.45 the quadratic function can be form

Quadratic Functions

This is a quadratic function.

When x is 3 this quadratic function is zero.

f(x)

=

x2 - 4x + 3

x = 3

f(3)

=

32 - 4 × 3 + 3

=

9 - 12 + 3

=

0

The Factor Theorem states that if a function of xis zero when x = 3, then (x - 3) is a factor of the function.

f(3) = 0 (x - 3) is a factor of f(x) = x2 - 4x + 3

The other factor can then be found to give the completely factorised form of the quadratic.

By multiplication it is found that the other factor is (x -1) and f(x) = (x - 3)(x -1)

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The Factor Theorem states that if f(a) = 0 then (x - a) is a factor of the function f(x).

(x - a) is a factor of f(x) because f(a) = 0

The process can be used in reverse to find a quadratic function with particular properties. Find the function with factors (x - 3) and (x + 5)

The function with factors (x -3) and (x + 5) is

f(x)

=

(x - 3)(x + 5)

=

x2 + 2x - 15

Sometimes functions with higher powers of x can be factorised by this method .

The process of obtaining the factors becomes longer and sometimes involves long-division.

f(x) = x3 - 12x2 + 47x - 60 f(5) = 0 (x - 5) is a factor.

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Long Division

Long division takes practise. Here is an example:

Divide: f(x) = x3 - 12x2 + 47x - 60 by the factor (x -5) to find the other factors

x cubed divided by x gives x squared multiply the factor (x -5) by x2 to give x3 - 5x2 subtract this from the first two terms and then divide the answer by x This gives -7x and after multiplication 7x2 + 47x Subtract to leave 12x which when divided by x gives 12

Long division is not essential. Other factors can be found using the Factor Theorem again. Factorising continues with the quadratic factor.

Therefore, the higher the R^2 (0?R^2?1), the closer the estimated regression equation fits the sample data. In this case, R^2 is 0.99624, which very close to unity. It indicates that the absolute equation fits the data very well according to R^2.

Some may use symmetry, some just trial and improvement, others may think about patterns of numbers. Extension work: Can the mean be 5? Can you get the mean to be between 6 and 7? * Obtain a mean of 5 and mode of 4 * Obtain a mean of 6 and mode of 3.

27=a5+b (2) 9=a (2)-(1) Substitute 'a' back into (1) 18=9x4+b 18=36+b b= -18 Substitute 'a' and 'b' back into second number equation S=9w-18 So we have the equations for the first and second numbers of the table. So now we put them together. The first number equation is solved by 4W-9 and then to that you had to times

Therefore a value added to x� translates the graph up or down on the y-axis. A different form of the quadratic function is to put it into a perfect square. A perfect square is an equation where the square root to the whole of one side is taken to get the answer on the other side.

(15X20-5X10)] We must then invert the corresponding signs of the integers using this shape: [ + - + ] [ - + - ] [ + - +] The final result is: [6 30 -120] [25 -40 -20 ] [-80 -40 250 ] We now know that A-1 is equal to the above result.

= x3-5x+0.1 f'(x) = 3x2-5 The iterative formula for the Newton-Raphson method is: xn+1 = xn- f(xn) f'(xn) My iterative formula is: xn+1 = xn- x3-5x+0.1 3x2-5 I will take my starting value of x (x1) to be 1.1. x1 = 1.1 x2 = -1.870073 x3 = -2.400053 x4 =

+ 6b = 30 6b = 30 - 21 6b = 9 b = 9/6 b = 3/2 Substitute a = 1/4, b = 3/2 into equation 12. I am going to substitute the value of a= 1/4 and b= 3/2 into equation 1, so I could find the value of c.

5 z4 = 52 + 1 = 26 and we see that this orbit tends to infinity, therefore 1 is not included in the Mandelbrot Set. If, however, the z0 valued 0 and the seed was 0, then the orbit would remain in a fixed position.