Maths

The value of a quantity is its magnitude expressed as the product of a number and a unit, and the number multiplying the unit is the numerical value of the quantity expressed in that unit.

More formally, the value of quantity A can be written as A ={A}[A], where {A} is the numerical value of A when the value of A is expressed in the unit [A]. The numerical value can therefore be written as {A} = A / [A], which is a convenient form for use in figures and tables. Thus, to eliminate the possibility of misunderstanding, an axis of a graph or the heading of a column of a table can be labeled “t/°C” instead of “t (°C)” or “Temperature (°C).” Similarly, an axis or column heading can be

labeled “E/(V/m)” instead of “E (V/m)” or “Electric field strength (V/m).”

Example:

In the SI, the value of the velocity of light in vacuum is c = 299 792 458 m/s exactly. The number 299 792 458 is the numerical value of c when c is expressed in the unit m/s, and equals c/(m/s).

Space between numerical value and unit symbol In the expression for the value of a quantity, the unit symbol is placed after the numerical value and a space is left between the numerical value and the unit symbol. The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle: °, ', and ", respectively, in which case no space is left between the numerical value and the unit symbol.

Example: = 30°22'8"

This rule means that: The symbol °C for the degree Celsius is preceded by a space when one expresses the values of Celsius temperatures.

Example: t = 30.2 °C but not: t = 30.2°C or t = 30.2° C

Number of units per value of a quantity

The value of a quantity is expressed using no more than one unit.

Example: l = 10.234 m but not: l = 10 m 23 cm 4 mm

Symbols for numbers and units versus spelled-out names of numbers and units.

This guide promotes using the symbols for the units, not the spelled-out names of the units. This will allow understanding by as broad an audience as possible, including readers with limited knowledge of English.

Example:

the length of the laser is 5 m but not the length of the laser is five meters

Choosing SI prefixes

The selection of the appropriate decimal multiple or submultiple of a unit for expressing the value of a quantity, and thus the choice of SI prefix, is governed by several factors.

These include:

• the need to indicate which digits of a numerical value are significant,

• the need to have numerical values that are easily understood,

• the practice in a particular field of science or technology.

A digit is significant if it is required to express the numerical value of a quantity. In the expression l = 1200 m, it is not possible to tell whether the last two zeroes are significant or only indicate the magnitude of the numerical value of l. However, in the expression l = 1.200 km, which uses the SI prefix symbol for 103 (kilo, symbol k), the two zeroes are assumed to be significant because if they were not, the value of l would have been written l = 1.2 km.

It is often recommended that, for ease of understanding, prefix symbols should be chosen in such a way that numerical values are between 0.1 and 1000, and that only prefix symbols that represent

the number 10 raised to a power that is a multiple of 3 should be used.

Examples:

3.3 x 107 Hz may be written as 33 x 106 Hz = 33 MHz

2703 W may be written as 2.703 x 103 W = 2.703 kW

5.8 x 10-8 m may be written as 58 x 10-9 m = 58 nm

Roman numerals

It is unacceptable to use Roman numerals to express the values of quantities.

Proper names of quotient quantities

Derived quantities formed from other quantities by division are written using the words “divided by” or “per” rather than the words “per unit” in order to avoid the appearance of associating a particular unit with the derived quantity.

Example: pressure is force divided by area or pressure is force per area but not: pressure is force per unit area

Dimension of a quantity

Any SI derived quantity Q can be expressed in terms of the SI base quantities length (l) , mass (m), time (t), electric current (l) , thermodynamic temperature (T) , amount of substance (n), and

luminous intensity (Iv)

A derived quantity of dimension one, which is sometimes called a “dimensionless quantity,” is one for which all of the dimensional exponents are zero: dim Q = 1. It therefore follows that the derived unit for such a quantity is also the number one, symbol 1, which is sometimes called a “dimensionless derived unit.”

Fibonacci sequence is a set of numbers based on adding the previous 2 elements in the sequence together

1, 1, 2, 3, 5, 8, 11, 19, 30, 49, 79 .....

The sequence has been much described as the maths of nature as many natural occurrences follow the sequence. For evidence of this in nature http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

Some more here on a young student that used it to build a better solar panels http://www.amnh.org/nationalcenter/youngnaturalistawards/2011/aidan.html

powers

Any number to the power of 0 is 1

x0 = 1

try to look at any integer and power it up to a factor of 5, 4, 3, 2, 1 ...... what is happening each time to the answer ...... and now for zero ??

Next proof .... use the law of indices to show how any number to the power of something is not changed by mulitplying or dividing by the number to the power of zero.

Calculus

Differentiation from 1st Prinicipals

f(x) = some function of x

to x add h .... thus find f(x+h)

calculate the entire equation

subtract f(x) to find f(h)

Divide this equation by h

h is a small quantity and thus it tends to →0 ... thus replace all h with 0 ...

gather terms