Maths for 'Behind the Tapestry: The Threads Revealed'

This page summarises the concepts you need to be happy with if you want to understand the supporting maths presented in 'Behind the Tapestry'.  No guarantees are offered, but if you can follow these (or alternatively you're prepared to take them on trust) then it's unlikely you'll meet anything in the book that will throw you.  You are reminded of all of these in the text of the book.

The main question is: are you frightened by algebra?  If you are then this book is not for you.  If you're not, then pretty well everything you need to know is included in the text – almost nothing is assumed except that you're willing to give it a go.

1.

Pythagoras' Theorem

a2 = b2 + c2  , where a is the hypotenuse (longest side) of a right angled triangle and b & c are the other two sides.

2.

sin2 x + cos2 x = 1

and

tan x = sin x / cos x

where x is any angle.  This follows directly from Pythagoras' Theorem and the definitions of sine x (opposite/hypotenuse) and cosine x (adjacent/hypotenuse).

This follows directly from the definition of tangent x (opposite/adjacent).

3.

Cosine Rule

a2 = b2 + c2 – 2bc cos A  , where a, b and c are the three sides of any triangle and A is the angle in the triangle opposite side a.  (Pythagoras' Theorem is a special case of this, where angle A is 90 degrees and so cos A = 0  - so the last term disappears.)

4.

Cancelling

You need to understand cancelling of algebraic (letter) terms on the top and bottom of an algebraic fraction, including brackets.

5.

Approximation

If c is very big compared to v and w, then (c2 – w2)/(c2 – v2) is approximately equal to c2/c2, which is equal to 1.

6.

speed, time, distance

speed = distance/time,  distance = speed x time,  time = distance/speed

7.

sine of (x – y)

sin (x – y) = sin x cos y – sin y cos x

8.

Solution of a quadratic equation

You need to know the formula for the solution of a quadratic equation (loads of entries for it on Google) - or be prepared take the result on trust.

9.

Indices

You need to know that power ½ is the same as a square root and that a power on the bottom half (denominator) of a fraction is the same as a negative power.  That's all shown in detail in the book.  Also that (for example) z3/2 = z1½ = z x sqrt(z)

10.

Differentiation

There is ONE differentiation in this book.  Another 'differentiation' consists of simply removing the integral sign from an integration - as differentiation is just the opposite of integration (so that doesn't need to be done either).

The ONE differentiation is to find the differential (d/dv) of:

E0 / (1 – v2/c2)½

(Where E0  is a constant anyway, so that stays as it is)

The book goes through it step by step, showing the parts that make up the result (including the differential of the term in the bracket).

There is also one use of the 'chain rule'.  That says, for example, that:

d/dt of (something) is the same as dx/dt x d/dx of (something).

This is effectively the opposite of cancelling - notice that the two dx's could be cancelled, getting us back to d/dt.

11.

Integration

There is NO integration needed in this book.  The concept of integration is used (and very thoroughly explained) - but ALL the integrals cancel or reduce to a trivially simple and very obvious result.

12.

Factorising

You need to be able to see how common factors (numbers and letters) are taken outside a bracket for a collection of terms.

In addition the equation of an ellipse is used, in terms of sines and cosines.  That's all laid out very clearly in the text and you don't need to have any previous knowledge on that subject in order to understand it thoroughly.

[back to info on book]