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a × b ÷ b = a 5 × 4 ÷ 4 = 5

Sometimes you will see a multiplication of letters as a dot or without any symbol. Don't worry, its exactly the same thing. Mathematicians areefficient and like to write things in the shortest, neatest way possible.

a b c = a × b × c a · b · c = a × b × c

It is usually neater to write known numbers to the left, and letters to the right. So although 4 x and x 4 are the same thing, it looks better to write 4 x . In this case, the “4” is a constant that is referred to as the coefficient of x .

Commutativity for multiplication

The fact that a b = b a is known as the commutative property of multiplication. Therefore, both addition and multiplication are described as commutative operations.

Brackets

Brackets Sometimes people say “parentheses” instead of “brackets”. in mathematics are used to show the order in which you must do things. This is important as you can get different answers depending on the order in which youdo things. For example:

( 5 × 5 ) + 20 = 45

whereas

5 × ( 5 + 20 ) = 125

If there are no brackets, you should always do multiplications and divisions first and then additions and subtractions Multiplying and dividing can be performed in any order as it doesn't matter. Likewise itdoesn't matter which order you do addition and subtraction. Just as long as you do any × ÷ before any + - . . You can always put your own brackets into equations using this rule to make things easier for yourself, for example:

a × b + c ÷ d = ( a × b ) + ( c ÷ d ) 5 × 5 + 20 ÷ 4 = ( 5 × 5 ) + ( 20 ÷ 4 )

If you see a multiplication outside a bracket like this

a ( b + c ) 3 ( 4 - 3 )

then it means you have to multiply each part inside the bracket by the number outside

a ( b + c ) = a b + a c 3 ( 4 - 3 ) = 3 × 4 - 3 × 3 = 12 - 9 = 3

unless you can simplify everything inside the bracket into a single term. In fact, in the above example, it would have been smarter to have done this

3 ( 4 - 3 ) = 3 × ( 1 ) = 3

It can happen with letters too

3 ( 4 a - 3 a ) = 3 × ( a ) = 3 a

Distributivity

The fact that a ( b + c ) = a b + a c is known as the distributive property.

If there are two brackets multiplied by each other, then you can do it one stepat a time:

( a + b ) ( c + d ) = a ( c + d ) + b ( c + d ) = a c + a d + b c + b d ( a + 3 ) ( 4 + d ) = a ( 4 + d ) + 3 ( 4 + d ) = 4 a + a d + 12 + 3 d

Negative numbers

What is a negative number?

Negative numbers can be very confusing to begin with, but there is nothing to be afraid of. The numbers that are used most often are greater than zero. Thesenumbers are known as positive numbers .

A negative number is a number that is less than zero. So, if we were to take a positive number a and subtract it from zero, the answer would be the negative of a .

0 - a = - a

On a number line, a negative number appears to the left of zero and a positive number appears to the right of zero.

On the number line, numbers increase towards the right and decrease towards the left. Positive numbers appear to the right of zero and negativenumbers appear to the left of zero.

Working with negative numbers

When you are adding a negative number, it is the same as subtracting that number if it were positive. Likewise, if you subtract a negative number, it is the sameas adding the number if it were positive. Numbers are either positive or negative and we call this their s ign. A positive number has a positive sign ( + ) and a negative number has a negative sign ( - ).

Questions & Answers

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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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