DIVISION BY ZERO AND INFINITY

Dividing any number by any random number will surely fetch you an easy result you are looking for but the scenario changes when you try to divide any number by zero or infinity.
Lets try to understand each case.

Any number divided by infinity is zero

First of all, what is infinity ∞? Infinity is a very huge number, a number larger than any assignable or countable number, Right? But can we come up with largest number ever discovered? No, I don't think so. There is always a bigger number than the other.

When we divide number by larger and larger number, the resultant value keeps getting smaller and smaller or we can say that the answer keeps coming closer to zero.
Divide 10 by X and keep increasing value of X

10 / 10 = 1
10 / 100 = .1
10 / 100000 = .0001
10 / 10000000000 = .000000001
10 / 100000000000000000000 = .0000000000000000001

Looking at the above pattern, we see that the answer keeps getting closer to zero as we increase the divisor.
So, for any huge hypothetical number defined as infinity as divisor, we assume that the answer is zero.

Let us understand this by a different logic.

If i ask you to divide 10 by 2, its easy you will say the answer is 5.
But what does it mean.
It means how many times can 2 be deducted from 10.

10 - 2 = 8          (1)
 8 - 2 = 6           (2)
 6 - 2 = 4           (3)
 4 - 2 = 2           (4)
 2 - 2 = 0           (5)

That means, if we deduct 2 five times from 10 we get 0.
what happens when we deduct 0 from 10.
Zero can be deducted countless times from 10 and this answer is called infinity.

So, 10 / 0 = ∞
Using above equation,
10 / ∞ = 0.

Any number divided by zero is infinity

Dividing any number by zero would give us infinity or undefined. OK, to make it simpler, if i tell you to handover 10 candies to 5 children in boxes before eating them. Its easy, you will place 2 candies in each box and give it to 5 children and eat the rest candies. If  i ask you to give 0 candies to 5 children in boxes and then eat the rest. You can still hand over empty boxes to them and the eat the rest.

0 / 6 = 0

But, what if i tell you to distribute these 10 candies in boxes to 0 children and then eat the rest. Now, there is no way you can do it. You will keep on sitting there till eternity trying to distribute those 10 candies to 0 children and you will never be fortunate enough to let those candies go in your mouth. Empty boxes can be given to children but candies cannot be given to nobody.
Here, candies is divident and children is divisor if you observe.

so 10 / 0 remains undefined or ∞

Mathematically,
Let us consider division of 10 by X where X  will keep becoming a smaller and smaller number approaching 0.

10 / 10 = 1
10 / 1 = 10
10 / .0001 = 100,000
10 / .0000000001 = 100,000,000,000
10 / .0000000000000000001 = 100,000,000,000,000,000,000

So, we see that as the divisor is getting smaller and smaller or approaching zero, the answer keeps getting huge and huge.

So,
X / 0 = ∞

Inversely,
X / ∞ = 0

And

0 / 0 = 0
∞ / ∞ = ∞








IVIDING

LEARNING WITH UNDERSTANDING : FEYNMAN TECHNIQUE

Forgetting what we read and learn is a common problem. Lack of understanding plays a significant role in this. Often I also face this problem.  I study some topic in the morning to teach my kid and by the time he is back from school,  half of the content is washed away from my brain. To solve this problem,  I decided to Google and believe me what I found was amazing and worth trying.

Richard Feynman

I came across a Mental Model called Feynman Technique named after Richard Feynman, a Nobel Prize winning Physicist. He won Nobel Prize in 1965 for his work on Quantum Electrodynamics. Along with Scientist, he was a great teacher, a great explainer. Infact, the nickname given to him was the " Great Explainer". He put incredibly difficult concepts into so easy language which other people could understand.

Feynman says that if you want to understand a concept well, explain it. Most effective way of understanding a concept is to teach someone else what you want to learn. Explaining a concept works to improve your knowledge of concept. This Feynman technique is a specialised technique designed to help us learn anything and study more efficiently with good understanding of the concepts. It works for all the fields be it computer, maths,  astronomy, physics, chemistry, psychology etc. The basic idea of this technique is " YOU MUST NOT FOOL YOURSELF AND YOU ARE THE EASIEST PERSON TO FOOL. "

Feynman explained his technique in four easy steps:

1. WRITE DOWN THE CONCEPT AT THE TOP

Take a piece of paper and write down the title at the top. Write down everything you know about the topic on a page and add to that page everything you learn something new about it.

2. EXPLAIN THE CONCEPT

Explain the concept in simple plain english as if you are teaching someone else. Work through examples and see if you are able to explain them relevantly.

3. IDENTIFY PROBLEM AREAS

Identify problem areas then go back to the source of review or source material and try to understand these areas as solid as you understand the other areas. The gaps in your knowledge should be obvious. Revisit problem areas until you can explain topic fully.

4. PINPOINT COMPLICATED TERMS AND SIMPLIFY

Finally,  look at your explaination and pinpoint complicated terms and challenge yourself to simplify them in easy to understand words.

Using this technique is very helpful as:

1. It quickly overviews the concept and tells you where your knowledge is solid.

2. Instantly pinpoint the areas where you are shaky.

Remember, the key here is simplicity. How can i explain to a kid as the kid will ask "why,  why does that work?" Explaining to someone who doesnot have the base knowledge as you have is your ultimate test of understanding a concept. If you are unable to explain in simple terms, it simply means that we donot understand it.

After these four steps, you will be able to understand the concept more deeply and retention of knowledge would be much better.

BLACK HOLES - THE HUNGRIEST OBJECTS IN SPACE

Ask your kid to vaccum clean his room. What happens when he vaccum cleans his room. All the dust is sucked by a vaccum cleaner. A Black Hole is just like a giant vaccum cleaner cleaning the debris left behind in the outer space. But it is not the suction power that makes things fall into Black Hole instead it is some other force - the force of gravity. Let us understand this in detail.

A Black Hole is one of the strangest objects in space. It is an area in space where gravity is so strong that whatever comes under the influence of its gravity, will fall into it. Even light cannot escape from it. Because light cannot escape from it, Black Holes appear black. We know that light can travel faster than anything at a speed of 300,000 km/s. Imagine if light cannot escape from a Black Hole, nothing else that we know of can.

The term Black Hole is used because these giant objects look like black holes in space since they put out no light.

A Black Hole is not really a hole and it is not empty. It doesnot have a surface like planet or star. It is a region in space filled with a lot of material crammed into an extremely small space. Imagine your whole house stuffed in a small box. It is because of this that the Black Hole has a super strong gravity and this gravitational pull is so great that even light cannot escape.


FORMATION

Black Holes are formed from stars. When a massive star runs out of fuel (Hydrogen), and it no longer can support its weight, it explodes. This explosion is called Supernova. Then, due to pressure the star starts getting smaller and smaller squashing all its material. It becomes even smaller than atom. Imagine the entire star squashed into such small space. Due to its small size, enormous mass and huge density, the gravity is so strong that it will absorb light and become a Black Hole. Black Holes are formed only from a heavyweight star. These Black Holes can grow incredibly huge as they continue to absorb light and mass around them. They can even eat other stars surrounding them. So these Black Holes are the hungriest objects in space. Many scientist think that there are super-massive Black Holes at the center of the universe.


Can we see a Black Hole? 

Since Black Holes don't reflect any light, we cannot see them. We can see the effect they have on the area of surface around them. By extremely high gravity, they pull in surrounding material at a very high speed causing this material to become very hot and emit X-rays. These rays help scientist find them.


Structure of Black Hole

Black Holes are made of three main parts:-

- Outer layer called outer event horizon. 
We can escape from this zone as gravity is not as strong here.

- Middle layer called inner event horizon.
Here, there is no chance of escaping. The gravity is much stronger. So, anything that enters here gets captured and the objects start to fall towards the center of Black Hole.

- The center of Black Hole called singularity.
Here the gravity is strongest.


Types

Black Holes are of different sizes. Many are massive than our sun. Scientist think that the Black Holes are of two types:-

1) Stellar-mass Black Hole 

Imagine a black round object 3 - 15 times bigger than our sun. Stellar type Black Hole is made when gigantic star reaches old age with all its fuel finished causing it to explode. This explosion is called Supernova. The material left from Supernova collapses in on itself transforming into a very compact object. This is how Stellar-mass Black Hole is made. These are found to be scattered in the galaxies.

2) Super-massive Black Hole

It is made when hundred million suns are all squashed into one place. As its name suggests, these are way too massive as compared to Stellar-mass. Actual formation of these type is still an open field of research. These Black Holes are known to occur in the center of almost all large galaxies known to us.


Is there a Black Hole at the center of our galaxy? 

According to scientists, there is one super-massive Black Hole in the middle of our galaxy - the Milky way. It has the mass of about 3 million suns and is very far from Earth. Huge Black Holes are thought to occur naturally at the center of most galaxies and many have already been detected.


Are there many Black Holes in the galaxy?

There can be many Black Holes in a galaxy but most galaxies of the size of aur milky way or larger are thought to have gigantic Black Holes weighing millions or billions of times as much as sun at their centers. When galaxies merge,  these Black Holes may spiral in towards one another and collide.

In recent studies,  an international team which is lead by Stefanie Komossa of Germany has identified the first known case of super massive Black Hole flung from its host galaxy.

Interesting Facts:-

- Black Holes have the mass of several million suns.

- Black Holes differ from each other in mass and spin.

- They don't  live forever, but slowly evaporate returning their energy to the universe .

- Black Holes are often surrounded by a ring of debris.

- Idea of Black Holes was first proposed by two different scientists in 18th century. John Michell and Pierre-Simone Laplace.  In 1967, a physicist named John Wheeler coined the term Black Hole.

- Black Hole can suck up earth but scientist say that the nearest Black Hole which lies at the center of our galaxy is so far away from earth that it's virtually impossible.

- In December 2011, astronomers noticed that our galaxy's Black Hole has set its eyes on a nearby cloud of dust and gas. Scientists who have been watching this cloud for several years with huge telescopes in Chile, South Africa have noticed that not only is this cloud is moving in the direction of Black Hole,  but it has also doubled its speed reaching more than 8 million km/hr. In couple of years,  this super-massive Black Hole will get its meal.

MOON - ALL YOU WANTED TO KNOW

Moon is first best friend of kids. I myself remember talking to Moon when i was a kid. Why moon looks so alluring to kids? Maybe because of its colour and shine.
When we gaze at the night sky, the first thing that attracts our attention is Moon. My kid often asks me that if moon is hung with some string. Thats funny indeed. But that is his imagination. Like him, many kids would be thinking in the same way. This blog is for those mothers who want to slowly introduce their kids to the world of science and reality.

All the bodies that we can see and that we cannot see in the sky are called celestial bodies or to be more precise, celestial bodies are the natural bodies outside earth's atmosphere. So now u know that moon is also a celestial body. Other examples are Sun, Stars, Planets etc. Moon is a Natural Satellite, i.e. it is a space body which orbits a planet. All natural satellites are called moon. Our planet earth has one natural satellite, i.e. the moon that we see in the sky. It orbits our planet and hence is the only natural satellite of earth. Satellites don't have their own light. They reflect Sun's light and hence appear shiny to us. If Sun wasn't there we would not be able to see the moon. The sun always illuminates one side of the moon.

AGE

Moon is almost 4.5 billion years old and formed more than 30 million years after the earth was formed. It is about 250,000 miles away from earth.

FORMATION

A large space body hit Earth, cutting off rocks from its surface. These rocks started revolving around earth due to earth's gravity and in due course of time, these rocks came together to form a big melted mass. This mass started to cool down over the period and hence moon was formed.  For another 500 million years,  pieces of rocks kept striking against the surface of moon causing damage in form of scars, craters, pits which can be seen using a telescope.

REVOLUTION AROUND THE EARTH

Moon takes 27 days 7 hours and 43 minutes to revolve around earth. Moon's orbit is elliptical. Looking down from North pole, we would see the moon orbiting counter clockwise from west to east.

WHY MOON HAS DIFFERENT SHAPES?

Each night we look at the moon, it appears of different shape. Some nights there is no moon at all. Its not moon which changes shape, it is how much part of moon we can see. Moon doesnot have its own light but it reflects sun's light and we see that side of moon which gets illuminated. If the entire side of moon that is lit up by the sun is facing us, we see a full moon. If only half of the moon's lit up side is facing us, we see a quarter moon. These changes in the size of the moon are called phases of the moon. It is actually the amount of moon that we see from earth depending upon how much of moon is lit up by the sun. The amount changes each day. As the moon moves around the earth, we can see different fractions of the illuminated moon. Each month our moon passes through 8 phases. These phases are named after how much moon we can see and whether the amount visible is increasing or decreasing each day.

LUNAR ECLIPSE

Eclipse occurs when light from sun is temporarily blocked off by earth or moon. There are two types of Eclipses
Solar Eclipse
Lunar Eclipse
Lunar eclipse occurs when moon travels behind earth. All moon, earth and sun comes in a straight line. In lunar eclipse,  earth comes between sun and moon.

FACTS TO KNOW

1. The first man to make proper map of moon was Galileo.

Galileo Galilei

2. The first spacecraft to actually orbit the moon was Luna 10 in 1966 launched by USSR.

3. First person to land on moon was Neil Armstrong in 1969. He said, " that's one small step for man, one giant leap for mankind ".
Till date only 12 people have been to moon and all are Americans.

4. Our moon is the 5th largest moon in solar system.

5. Moon has no atmosphere so no sound can travel.

6. We always see one side of the moon from earth. It is called near side while the other side is called far side.

7. Moon gravity is about 1/5 as compared to earth.

8. Moon is moving away from earth at about 3.8 cm every year.

9. Moon is much smaller than earth.  Its diameter is 2,159 miles and that of earth is 7,917 miles i.e. 4 moons can fit into earth.

10. Moon has no air, no water and no life.

11. Moon rises in the east and sets in the west.

12. Moon is very hot during during day and very cold during night.

13. Gravitational attraction  of moon and sun together cause tides on earth.

14. Eclipse of moon lasts for several hours while the moon passes through the earth's shadow.

15. On moon, we would weigh 1/6th of our weight on earth.

16. NASA wants to set up a permanent space station on moon.

MAGNETISM

Science is everywhere around us and understanding it makes our life much easier and fascinating too. Kids love to play with magnets and playing with magnets give initial lessons of science to them. Imagine how fascinating would it be for kids to see magnets attracting their toy cars,  paperclips, cells, cans etc. So exactly what is a magnet.

Lodestone

A magnet is something that attracts certain metals like iron,  nickel, cobalt. It has invisible force that pulls certain metals. This force is called MAGNETISM. A lodestone is a natural magnet.

Magnets that we see in the market come in different shapes like bar magnet,  horseshoe magnet, flat circular disc-shaped etc but common feature of all is that all magnets have two ends called poles. One is NORTH POLE and other SOUTH POLE.

When we hold two magnets together,  they either ATTRACT (pull) or REPEL (push away). To understand this, let us know more about magnetism.

Every magnet has a magnetic field around it which is the invisible lines of magnetic force which leave magnet from its north pole and re enters the magnet at its south pole.  We can call it, invisible area of magnetism .

On bringing two magnets closer,  the north pole of one magnet will attract south pole of other magnet and vice versa but two north poles and two south poles will always repel.
So,  we learn that LIKE POLES REPEL AND UNLIKE  POLES ATTRACT. 

A very significant feature of magnet is that its magnetic field is stronger at the poles than in the center. We can see this by immersing magnet in iron fillings,  almost all the pins will stick at the two ends of a magnet which shows that magnet is stronger at its poles or ends.

Magnets can be temporary or permanent. The magnets that we get in the market are permanent ones.  These donot lose their magnetic properties. Temporary magnets behave like magnet only if they are in contact with permanent magnet.
e.g. if a magnet is brought near iron nail, it will stick to magnet and behave like a magnet. It will now attract another nail which will stick to first nail and likewise a chain is formed as all nails will start behaving like a magnet but once the permanent magnet is removed,  they will become ordinary nails.

A bar magnet if suspended freely in air will always allign itself in a particular direction. The North pole of a magnet points towards earth's geographical North and South pole of a magnet points towards earth's  geographical South. This property is used in making Compass.

A Compass is made of tiny magnet in shape of needle which can spin. The North pole of compass needle will always point towards earth's geographical North pole. So it helped sailors find their way in the sea.

It is of great interest to know that our Earth behaves like a giant magnet. The core of the earth has molten rocks rich in iron responsible for this magnetic behaviour. This is the reason why a freely suspended magnet always align along the geographical N-S direction.

USES OF MAGNETS

1. Magnets are used to seperate magnetic substances from a mixture.
2. The doors of refrigerator has a weak magnetic strip to help keep the door closed firmly.  The doors of the cupboards have magnetic latches for the same purpose.
3. Magnets are used in microwaves, loud speakers, motors, generators etc.
4. Magnets find their use in computer hard disk,  floppies,  audio and video tapes.
5. Magnetic stripes on credit cards, ATM cards, airline tickets is used to store information.

IMPORTANT FACTS

1. Earth has magnetic field which stretch out into space and can affect things around.
2. Sun has magnetic field several times stronger than earth.
3. Moon has little or no magnetism.
4. Other planets like Saturn, Jupiter,  Neptune and Uranus have stronger magnetic field than Earth while Mars,  Mercury and Venus has weaker field.

VEDIC MATHS - NIKHILAM MULTIPLICATION

Nikhilam mulltiplication is a quick and easy method for multiplying two numbers which are both close to a power of 10, 100, 1000 etc. In this method, the calculations are made easier using a complement or deficiency. You can refer my earlier blog on complements.

To understand  this method, let us start with simple multiplication using base 10.

Base 10
multiplying below base 10

We know,  6 x 8 = 48

Lets do this by Nikhilam multiplication.
Here,  10 is the base as both 6 & 8 are nearest to 10.

Step 1
Find the complement  of 6 & 8.
Complement of 6 = 4
Complement of 8 = 2

Step 2
Write these complements on RHS with connecting minus sign.

      10

      6 - 4

   X 8 - 2

   ________

         /

Note:-

Minus sign shows that the complements ate both less than 10.

/ is used to distinguish two parts of the answer. 

Step 3

On the RHS, vertically multiply the two compliment digits and write it down

4 x 2 = 8

      10

      6 - 4

   X 8 - 2

    ________

          / 8

Step 4

For LHS,  cross subtract 

Either 6 - 2 or 8 - 4,

The result will be same i. e. 4

Write down on the left. 

      10

      6 - 4

    X8 - 2

__________

      4 / 8

So,  the final answer is 48

Try one more example

     7 x 9 

      10

      7 - 3

      9 - 1

   ________

      6 / 3

Multiplying  above base 10

When numbers are above base 10 then instead  of deficiencies, above the base has surpluses. So the subtraction part changes to addition.

i. e.  Cross add instead of cross subtract. 

 Let's  do 12 x 11

Step1

First set down the surpluses 02 and 01.

12 is 2 more than base 10

11 is 1 more than base 10

      12 + 02

   X 11 + 01

     __________

Step 2

Multiply 2 x 1 and write the answer on RHS. 

      12 + 02

    X11 + 01

    __________

            /  2

Step 3

Cross add 12 + 1 or 11 + 2 , answer is 13

Write on LHS

      12 + 02

    X11 + 01

     _________

       13 / 2

Answer is 132.

Ofcourse,  calculations  are quicker if we know tables but this method becomes highly useful in higher numbers having the base close to 100, 1000 and so on. 

      

   

SMALL QUESTIONS TO GREAT MINDS

I can tell from my experience that all moms come across two words everyday ' WHY and HOW' as our little ones have just been exposed to this mysterious world. For them, everything revolves around WHY and HOW. I am myself a mom of a 6 year old boy and when i try to see the world from his eyes, I realise that the busiest brains in this world are not of a scientist but that of a kid who is always struggling to solve the new building mysteries around him with each passing minute. A scientist questions the things unknown to him but to our little scientists, the whole universe is unknown or like some foreign land. Just imagine the load of bag full of questions that they carry in their head.To lighten this burden, the first one they look upto is their MOM- their first most loved and most trusted encyclopedia. No one teaches them but maybe by some divine power they already suppose that their mom knows everything and only she can unfold the world of mysteries building in their mind. This is the stage when critical thinking starts to develop in our kids.

If we ponder over those little questions popping up in our kids brains, they might seem very irrelevant to our evolved brains but these questions makes all the difference to our little ones and are in one way also building the personality of our kids. So think, Are they really irrelevant?

Here i would like to exemplify this approach. Whenever i take my kid out at night in the car, his question is " MAMA why moon follows us wherever we go?". Looking at his innocent eyes, I wonder if i can ever explain him this phenomenon as i think he is too small to understand this concept. This is our first mistake;

 'UNDERESTIMATING THE POTENTIAL OF OUR KIDS'.

Our kids have a lot of potential in them. Its our job to use it and nurture it furthur. Kids are capable of understanding a lot of things. Its we who have to learn to be good teachers i.e.  choosing the right and easy words which are understandable to them.

In the first place, kids always approach us with their queries. They always take the first step forward but we are too judgmental. Our common answer to their queries is " you are too small to understand this". I myself used these words and my child said, " mama, then make me understand". His reply changed my thinking and gave me a new vision.

We teach our kids to not to be lazy but in reality we have taught them to be so by actually showing laziness to solve their little queries. Each question they ask opens door to a new learning and by quashing these questions somewhere somehow we are closing their window of questions and new learning, thereby, in a way we are playing a vital role in hindering their brain development. I know its not very pleasant to read but unfortunately, its true.

Its our responsibility to not to give a deaf ear to these little questions as their relevance can be well established if we see the world from their window. Everything would be so mysterious and at the same time mesmerizing like Harry Potter's land. Its now that we can write anything on our kid's mind. Its now that their thinking process can be developed to a boundary-less thinking.

In my upcoming blog, i will talk about developing critical thinking in kids.

GRAVITY FOR KIDS

I am sure gravity is the best toy of toddlers as they love dropping everything down and asking mom to pick it up.  Ask any child about gravity and the answer would be " it makes everything fall". If i ask you to jump,  you will land on your feet.  But why would you land? It is because of gravity.  It pulls everything towards the ground.

Gravity has not been understood by our scientist completely.  But in simple terms,  gravity is invisible attractive force that causes objects to move towards each other.

Imagine if there was no gravity on earth. Water, animals,  trees,  you, me,  everything will fly off in space.

The first scientist to study gravity is Sir Issac Newton about 300 years ago. There is famous story that once Newton was sitting under apple tree thinking and suddenly apple fell on his head.  But why not it went up or sideways why only down. It was this thought that  led Newton to discover the force that made it occur and he called it Gravity.  

He discovered :

- What goes up must come down.

- Which ever direction the object is thrown, up, down, left or right, it must come           down.

- Gravity works the same on all objects in absence of wind resistance.

- Gravity always pulls,  it never pushes.

Apart from the small scale examples like jumping, throwing and falling of an apple, gravity is also important on a larger scale.  How?

It is the sun's  gravity that keeps planets orbiting around it.  So sun's gravity acts as a glue that binds solar system.

Important  facts:

1) Ocean tides are caused by gravitational  pull of moon on the ocean.

2) Black holes have strongest gravitational  pull in the entire universe.

3) Along with discovering gravity,  Sir Issac Newton also gave three laws of motion.

4) Gravity has no duality, it only attracts,  never repels.

5) To leave earth's  gravitational  pull behind,  an object must travel 17 miles per second i.e. our planets escape velocity.

6) Moon has less gravity than the earth. This means that the object weighs less on moon than on earth. The moon's gravity is 1/6th of the earth's gravity.

7) Gravity is by far the weakest of four fundamental  forces.

I hope this concept of gravity is now very simple to understand. 

Vedic maths : Complements

 COMPLEMENTS

A complement is a number which must be added to a number to make it to 10, 100, 1000 and so on. You can also call it a deficiency from a power of 10.

eg. Complement  of 6 for 10 is 4
       As 6+4=10
      Complement of 93 for 100 is 7
       As 93+7=100
      Complement of 992 for 1000 is 8
       As 992+8=1000

We can also say that
992 is the complement of 8 and 8 is the complement  of 992.
6 is the complement  of 4 and vice versa.

How  to find a complement

The easy way to find the complement of a number from 100 is to subtract first number from 9 and last number from 10.
eg. For 83,
       9-8=1
     10-3=7
So the complement of 83 from 100 is 17.

       For 74,
       9-7=2
     10-4=6
So 26 is the complement of 74 and 74 is the complement of 26.

If the number is more than 2 digits even then the same rule applies " all from 9 and last from10 ".

eg.  876
       9-8=1
       9-7=2
     10-6=4
So,  124 is the complement of 876.

See the number 3400, What do we do here?
In the cases like this when number ends in zero,  take the last number before zero and subtract from 10 and rest from 9.
For no 3400,
             9-3=6
           10-4=6
Two zeroes will be placed at end.
So the complement of 3400 is 6600 as 6600+3400=1000.

For 1030,
Complement is 8970
As, 9-1=8
       9-0=9
     10-3=7 followed by 0.

This concept of complements is used for nikhilam multiplication in vedic maths which will be discussed in detail in my next blog.
   

How to use Articles

Articles are A,  An and The.
The simplest key to use a and an is that a is used before nouns beginning with consonants .
eg.  Sam is reading a book.

an is used before nouns beginning  from vowels.
eg.  I am eating an orange.

Please note-
1. Some vowels have a consonant sound.  In cases like these a is used with nouns.
eg.  I've been to  a European  city many times.
2. Some words begin with H that we donot hear while saying the word.  In these cases,  an is used before  nouns .
eg.  I have been waiting for an hour. 

The is used before a noun when we are talking to someone who already understand which person or thing we mean. Intrestingly, use of a and the confuse people.  Let us understand this concept with examples .
Have you saved the money that i gave you?
My mother is working in the kitchen.

The simple examples above will help you understand the use of a, an and the.