## OPERATIONS ON NUMBERS

Addition of two or more numbers gives us another number. The numbers which are added are called addends and the new number thus obtained is called the sum. For example,
34670 + 12345 = 47015
Here, 34670 and 12345 are called addends and 47015 is called the sum of 34670 and 12345. Large numbers are added in the same way as the small numbers.

When we add three numbers, we can add any two numbers first and then add the third number to the sum. In other words, we can group any two of the three numbers in order to find the sum of the three numbers. If we have to add more than three numbers, we can similarly group any two of the given numbers in several ways and add them. For example, suppose we want to add 234523, 123098, 555623, and 876543. We can find the sum in any of the following ways:

• [(234523 + 123098) + 555623] + 876543
= (357621 + 555623) + 876543
= 913244 + 876543 = 1789787
• [(234523 + 123098) + 876543] + 555623
= (357621 + 876543) + 555623
= 1234164 + 555623 = 1789787

So, we can add three or more numbers by grouping them in any way we feel convenient.

### Subtraction

Subtraction of one number from another number gives us a third number. The new number thus obtained is called the difference of the two numbers. For example,
70000 – 67429 =2571
Here, 2571 is the difference of 70000 and 67429. Large numbers are subtracted in the same way as the small numbers.

### Multiplication

Multiplication of two or more numbers gives us another number. The new number thus obtained is called the product of those numbers. For example, 11 × 13 = 143.
Here, 143 is the product of 11 and 13. 11 and 13 are called the/actors of 143. Note that 1 is always a factor of any number.
Let us take another example, say, 855 × 73 = 62415. Here, 855 and 73 are factors of 62415, and 62415 is the product of 855 and 73.
Large numbers are multiplied in the same way as the small numbers.

### Multiplication using zeros

When we multiply a number by 10,100,1000, etc. we simply place those many zeros on the right of that number. For example, if we want to multiply 15 by 10, then the answer will be 150. Similarly, if we want to multiply 15 by 100, then the answer will be 1500, and so on. Zeros have been added on the right side of 15.

### Division

Division of a number by another number gives two new numbers—the quotient and the remainder. The number that is divided is called the dividend and the number that divides is called the divisor. For example,

Here, 7 is the divisor; 66 is the
dividend; 9 is the quotient, and 3 is the remainder.

Divide 69205 by 432 and find the quotient and remainder.

Here, the quotient is 160 and remainder is 85.

### Mixed operations involving +, –, ×, and ÷

Till now we have solved problems involving only one type of operation, that is, one of the following: addition, subtraction, division, and multiplication. But what do we do if a problem involves two or more operations together? Consider the following problem:
Example: Simplify: 16 – 6 + 2 – 3

In the 1st case, the answer is 9 and in the 2nd case, it is 5. We get different answers depending on the order in which the operations are carried out. But one of the two answers we got is wrong. In order to avoid this kind of ambiguity, an international convention has been accepted.

1. If any mathematical expression has symbols of addition and subtraction both, we first add and then subtract. For example, consider the following case: 16 – 3 + 4 – 5
= 20 – 3 – 5    (Addition: 16 + 4 = 20)
= 17 – 5            (Subtraction: 20 – 3 = 17)
= 12                  (Subtraction: 17 – 5 = 12)
2. If in three operations [(+, – , and × ) or (+, – , and ÷)], that is, besides addition and subtraction, a problem involves multiplication or division, we first multiply or divide, and then go for addition and subtraction respectively. For example, consider the following cases.
(a) Simplify: 7 + 3 × 4 – 3
In the above example, the three operations involved are +, – , and ×. To solve this problem, we first multiply the numbers, then we go for addition, and at the end we subtract.
7 + 3 × 4 – 3
= 7 + 12 – 3 (Multiplication: 3 × 4 = 12)
= 19 – 3        (Addition: 7 + 12 = 19)
= 16               (Subtraction: 19 – 3 = 16)
(b) Simplify: 16 – 6 ÷ 2 + 8
In the above example, the three operations involved are +, – , and ÷. To solve this problem, we first divide, then add, and at the end subtract the number. 16 – 6 ÷ 2 + 8
= 16 – 3 + 8  (Division: 6 ÷ 2 = 3)
= 24 – 3        (Addition: 16 + 8 = 24)
= 21               (Subtraction: 24 – 3 = 21)
3. When a problem involves all the operations namely, +, – , × , and ÷, then there is an agreed formula denoted by ‘DMAS’, which mathematicians follow. In ‘DMAS’, D stands for division, M for multiplication, A for addition, and S for subtraction. DMAS represents the order of operations. For example, consider the following cases.
(a) Simplify: 5 + 4 × 3 – 9 ÷ 3
In the above example, all the four operations are there, so we have to use DMAS rule, as shown below: 5 + 4 × 3 – 9 ÷ 3
= 5 + 4 × 3 – 3 (Division: 9 ÷ 3 = 3)
= 5 + 12 – 3       (Multiplication: 4 × 3 = 12)
= 17 – 3              (Addition: 5 + 12 = 17)
= 14                     (Subtraction: 17 – 3 = 14)
(b) Simplify: 7 × 3 – 4 + 60 ÷ 10
In this example too, all the four operations are there, hence to simplify this we have to use DMAS rule. 7 × 3 – 4 + 60 ÷ 10
= 7 × 3 – 4 + 6 (Division: 60 ÷ 10 = 6)
= 21 – 4 + 6      (Multiplication: 7 × 3 = 21)
= 27 – 4             (Addition: 21 + 6 = 27)
= 23                    (Subtraction: 27 – 4 = 3)

### Using the operation ‘Of’

Sometimes we need to find the value of ‘$$\frac { 1 }{ 2 }$$ of 16’ or ‘3 of 5’.
This means we need to find the value of $$\frac { 1 }{ 2 }$$ × 16 or 3 × 5.
So ‘of’ means multiplication.
Hence, $$\frac { 1 }{ 2 }$$ of 16 = $$\frac { 1 }{ 2 }$$ × 16
= 8
and 3 of 5 =3 × 5
= 15.
When the operation ‘of’ appears in any mathematical expression, then it must be performed before any other operation. To solve such kind of expression we use ODMAS rule, in which O stands for of, D for division, M for multiplication, A for addition, and S for subtraction.
Consider the following examples.

Example 1: Simplify 36 ÷ 2 of 3 + 6 × 2.
Solution: To solve this, we first solve the operation ‘of’.
36 ÷ 2 of 3 + 6 × 2
= 36 ÷ 6 + 6 × 2  (Of. 2 of 3 = 2 × 3 = 6)
= 6 + 6 × 2           (Division: 36 ÷ 6 = 6)
= 6 + 12                (Multiplication: 6 × 2 = 12)
= 18                       (Addition: 6 + 12 = 18)

Example 2: Simplify 42 ÷ 6 × 2 + $$\frac { 1 }{ 7 }$$ of 35 × 2.
Solution: 42 ÷ 6 × 2 + $$\frac { 1 }{ 7 }$$ of 35 × 2
= 42 ÷ 6 × 2 + 5 × 2  (Of: $$\frac { 1 }{ 7 }$$ of 35 = $$\frac { 1 }{ 7 }$$ × 35 = 5)
= 7 × 2 + 5 × 2           (Division: 42 ÷ 6 = 7)
= 14 + 10                    (Multiplication: 7 × 2 = 14 and 5 × 2 = 10)
= 24                            (Addition: 14 + 10 = 24)

### Use of brackets and the BODMAS rule

Let us consider an example to illustrate the use of brackets.
Rima bought 35 chocolates and ate 5 of them. She distributed the remaining chocolates equally among 6 of her friends. How many chocolates did she give to each of them?

In this problem we have to subtract 5 chocolates that Rima ate, from 35 chocolates she had, before dividing them among 6 of her friends. So we have to first perform the operation of subtraction and then do division. In such cases, we use brackets around the part that has to be done first, that is
(35 – 5) ÷ 6 (First solve bracket, i.e., 35 – 5 = 30)
= 30 ÷ 6      (Division: 30 ÷ 6 = 5)
= 5
Consider another example.

Example 3: Solve 2 of 3 × (5 + 2).
Solution: 2 of 3 × (5 + 2)
= 2 of 3 × 7  (First bracket: 5 + 2 = 7)
= 6 × 7          (Of: 2 of 3 = 2 × 3 = 6)
= 42              (Multiplication: 6 × 7 = 42)
Hence, when problems involve brackets, of, ×, ÷, +, and – then
To make it easy to remember this order, we remember the word BODMAS, where B stands for brackets, O for of’, D for division, M for multiplication, A for addition, and S for subtraction. This is called the ‘BODMAS‘ rule.
Sometimes numerical expressions may involve different types of brackets. These brackets are

• Vinculum or bar —
• Parentheses or small brackets ( )
• Braces or curly brackets { }
• Square brackets or big brackets [ ]

We simplify expressions by starting with the innermost bracket. Usually the vinculum is the innermost bracket, next is the parentheses, then the braces, and finally the square brackets. Let us now consider some examples.

Example 4: Simplify $$25-[20-\{10-(7-\overline{5-3})\}]$$.
Solution:

RULES FOR SIMPLIFICATION

1. Order of operation: The use of brackets take us to a new order of operation. The operation inside the brackets comes before the ODMAS. There are different types of brackets already mentioned here.

2. If there is no sign between a number and the bracket, then it is implied that the operation to be performed is multiplication.
Examples

3. When there is ‘+’ sign before a bracket, you can simply remove the bracket.

Examples:

4. When there is a ‘–’ sign before a bracket, then all signs within the bracket change while removing the bracket.

Examples:

## Chemical Properties of Carbon Compounds

Though we have millions of organic compounds, the reactions they undergo are limited. Some important reactions among them are discussed here.
They are:
1) Combustion reactions
2) Oxidation reactions
4) Substitution reactions

1. Combustion reactions
Carbon, and its compounds burn in presence of oxygen or air to give CO2, heat and light.
The process of burning of carbon or carbon compound in excess of oxygen to give heat and light is called the combustion reaction. In the reactions carbon is in its maximum oxidation state of 4+ in the product.
Eg: 1) C + O2  →  CO2 + Energy
2) 2C2H6 + 7O2  →  4CO2 + 6H2O + Energy
3) CH3CH2OH + 3O2  →  CO2 + 3H2O + Energy
Generally, saturated hydrocarbons burn with a clear light blue flame, whereas unsaturated hydrocarbons burn with yellow flame with soot(carbon). If air is not sufficiently available during combustion, even saturated hydrocarbons give sooty flame. When Coal, Petroleum etc., burn in air they give oxides of sulphur and nitrogen in addition to CO2 and H2O which pollute the environment. When Coal or Charcoal is burnt sometimes they just glow red without flame. This is because to get a flame gaseous
fuels should burn.
Most of the aromatic compounds burn with sooty flame.
Because of the inlets of air getting closed, the fuel gases donot completely undergo combustion. Hence, it forms a sooty carbon form which gets coated over the vessels.A combustion reaction is generally defined as any reaction that sustains a flame. It usually involves burning with oxygen, though some exceptions
are there, combustion reaction is always exothermic, that is energy is liberated during combustion reaction.

2. Oxidation reactions
Though combustion is generally oxidation reaction, all oxidation reactions are not combustion reactions. Oxidation reactions may be carried out using oxidizing agents. Oxidizing agents or Oxidants are substances that oxidize other substances. They themselves undergo reduction.
Eg: Alkaline Potassium permanganate or Acidified Potassium dichromate in solutions act as oxidizing agents and supplies oxygen to convert alcohols into carboxylic acids. Ethyl alcohol undergoes oxidation to form the product Acetaldehyde and finally Acetic acid.(see following equation).

Unsaturated organic compounds that contain multiple bonds (=, ≡ bonds) like alkenes and alkynes undergo addition reactions to become saturated. During the reactions addition of the reagent takes place at the double bonded or triple bonded carbon atoms.

In the above reactions ‘Ni’ acts as ‘catalyst.’
Catalyst: A catalyst is a substance which regulates (increase/decrease) the rate of a given reaction without itself finally undergoing any chemical change.
These reactions are commonly used in the hydrogenation of vegetable oils using nickel as catalyst. Vegetable oils generally have long unsaturated carbon chains, while animal fats have saturated carbon chains.
Fats and oils are both of fatty acids. Oils are generally liquids at room temperature due to unsaturated fatty acids but fats are solids due to saturated fatty acids.

4. Substitution reactions
A reaction in which an atom or a group of atoms in a given compound is replaced by other atom or group of atoms is called a substitution reaction.
Alkanes, the saturated hydrocarbons are chemically least reactive.
Therefore they are also called paraffins (parum = little; affins = affinity,  i.e., no affinity towards chemical changes). However they undergo some chemical changes under suitable conditions which are substitution reactions.
For example, methane (CH4) reacts with chlorine in the presence of sunlight. Hydrogen atoms of CH4 are replaced by chlorine atoms.