The weird number trick

Keep scrolling down slowly... Just keep saying this (in your head, "five five five, six six six".: 555 666 555 666 555 666 555 666 555 666 555 666 555 666555 666555 666555 666555 666555 666555 666555 666555 666555 666555 666555 666555 666555 666555 666 555 666555 666555 666555 666555 666 555 666 555 666 now... THINK OF A NUMBER!!!!!!!!!!!!!!!! You thought of 76 didn't you?

Awesome Mind Trick

This little Jedi mind trick is kinda freaky, till you think about it a little while. Then it's even more weird. Just follow the instructions below:
DON'T scroll down too fast, do it slowly and follow the instructions below exactly, do the math in your head as fast as you can. It may help to say the answers aloud quietly.
FOLLOW these instructions one at a time and as QUICKLY as you can!












What is:


* 2+2?
* 4+4?
* 8+8?
* 16+16?













Quick! Pick a number between 12 and 5.
























Got it?


The number you picked was 7.







Ready for another?
Just follow these instructions, and answer the questions one at a time and as quickly as you can! Don't advance until you've done each of them. Now, ARROW down, but not too fast, you might miss something.........


What is:


* 1+5
* 2+4
* 3+3
* 4+2
* 5+1













Now repeat saying the number 6 to yourself as fast as you can for 10 seconds. Then scroll down.
























QUICK!!! THINK OF A VEGETABLE! Then arrow down.
























You're thinking of a carrot.
If not....
you're among the 2% of the population whose minds are warped enough to think of something else. 98% of people will answer 'carrot' when given this exercise.

One more added::::

Think a number quickly between 50 and 100 and it must have even and disticnt digits....
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I hope U thought of 68...

Math Magic/Tricks [VERY INTERESTING]

Trick 1: Number below 10
Step1: Think of a number below 10.
Step2: Double the number you have thought.
Step3: Add 6 with the getting result.
Step4: Half the answer, that is divide it by 2.
Step5: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.
Answer: 3


Trick 2: Any Number
Step1: Think of any number.
Step2: Subtract the number you have thought with 1.
Step3: Multiply the result with 3.
Step4: Add 12 with the result.
Step5: Divide the answer by 3.
Step6: Add 5 with the answer.
Step7: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.
Answer: 8


Trick 3: Any Number
Step1: Think of any number.
Step2: Multiply the number you have thought with 3.
Step3: Add 45 with the result.
Step4: Double the result.
Step5: Divide the answer by 6.
Step6: Take away the number you have thought from the answer, that is, subtract the answer from the number you have thought.
Answer: 15


Trick 4: Same 3 Digit Number
Step1: Think of any 3 digit number, but each of the digits must be the same as. Ex: 333, 666.
Step2: Add up the digits.
Step3: Divide the 3 digit number with the digits added up.
Answer: 37


Trick 5: 2 Single Digit Numbers
Step1: Think of 2 single digit numbers.
Step2: Take any one of the number among them and double it.
Step3: Add 5 with the result.
Step4: Multiply the result with 5.
Step5: Add the second number to the answer.
Step6: Subtract the answer with 4.
Step7: Subtract the answer again with 21.
Answer: 2 Single Digit Numbers.


Trick 6: 1, 2, 4, 5, 7, 8
Step1: Choose a number from 1 to 6.
Step2: Multiply the number with 9.
Step3: Multiply the result with 111.
Step4: Multiply the result by 1001.
Step5: Divide the answer by 7.
Answer: All the above numbers will be present.


Trick 7: 1089
Step1: Think of a 3 digit number.
Step2: Arrange the number in descending order.
Step3: Reverse the number and subtract it with the result.
Step4: Remember it and reverse the answer mentally.
Step5: Add it with the result, you have got.
Answer: 1089


Trick 8: x7x11x13
Step1: Think of a 3 digit number.
Step2: Multiply it with x7x11x13.
Ex: Number: 456, Answer: 456456


Trick 9: x3x7x13x37
Step1: Think of a 2 digit number.
Step2: Multiply it with x3x7x13x37.
Ex: Number: 45, Answer: 454545


Trick 10: 9091
Step1: Think of a 5 digit number.
Step2: Multiply it with 11.
Step3: Multiply it with 9091.
Ex: Number: 12345, Answer: 1234512345

Difference between pressure and stress

Pressure is the force per unit area. Though the dimensions of pressure and stress are the same , they are not the same quantity.

When the whole surface of a body's is acted upon by the forces, acting perpendicularly everywhere on it, the force per unit area is called Pressure. (see below fig)


Stress is also a force per unit area but it can be different on different surfaces. Also it is not necessary that the force should be perpendicular to the surface. For example, there is a stress on the cross section of a bar,( shown in below fig) but there is no stress on its sides.

Kepler's Laws

First law:

"The orbits of planets are elliptical with the sun at one of their two foci"

The elliptical path of a planet around the sun is shown in the figure. Here f1 and f2 are the foci of an ellipse and sun is at either of them.

Second law:

"The area swept by a line, joining the sun to a planet, per unit time is constant."

Third law:

"The Square of the periodic time (T) of any planet is directly proportional to the cube of the semi-major axis (a) of its elliptical orbit."

Practical Illustration to understand Newtons third law of motion

When the driver of horse cart, Mr. SAM whipped the horse named chetak, in order to make a move; chetak(the horse) looked back at Sam and said, "Sam, it seems you do not know Newton's third law of motion. According to this law, with whatever force F₁ Would i pull the cart forward, the cart would also pull me backward with the force F₂ and these two forces are equal in magnitude and hence the resultant of them would be zero, so there is no way can i make the cart move."

Chetak knew only that action and reaction are equal and opposite. But since Sam had secured 100/100 marks in physics, he very well knew that action and reaction act on different bodies. So he patiently explained to Chetak about forces which is as under....

Suppose the horse pulls the cart by a force F₁ in the forward direction and the cart exerts the force F₂ on the horse in the backward direction Here F₁ =F₂


We should first decide the system. If we take the horse as our system then we should consider only those forces which are acting on the horse. Since force f1 is on the cart we do not consider it but we have to consider the force f2 which acts on the horse. Then a question arises that why the horse goes forward when the force f2 on it is acting backward? The answer to this question is that still we have not considered all the forces acting on the horse.

In these process three action-reaction pairs are there...

1. a pair of forces exerted by the horse on the cart and that by the cart on the horse.
2. a pair of forces exerted by the horse on the ground and that by the ground on the horse.
3. a pairs of forces exerted by the cart on the ground and that by the ground on the cart.

When a horse walks, it pushes the ground and as a result the ground exerts force on the horse in the opposite direction. The horse accelerates forward if the forward components exceeds the force. Thus, the acceleration of the horse and that of the cart are equal in magnitude and hence they move together.

Fundamental Forces in Nature

Starting from the rudimentary concepts of force we have framed the scientific concept of force. Even the explanation given by the Aristotle has proven to be faulty. It was famous physicist Issac Newton who first gave the clear concept of force in his three laws of motions. Newton also gave the universal law of gravitation.

Besides gravitational forces we come across many different types of forces such as the frictional force between two surface, the restoring force arising in compressed springs, tension produced in a stretched string, force of surface tension prevailing in the free surface of liquid, viscous force in fluid medium, intermolecular forces etc. Magnetic and electric forces are the origin of all these forces.

Four fundamental forces (interactions):

1. The Gravitational force:

• The gravitational force is the force of mutual attraction between any two objects by virtue of their masses.


• It is a universal force.


• According to Newton's law of gravitation, this mutual attractive force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them(inverse square law).


• It does not require any intervening medium.


• compare to other fundamental forces gravity is the weakest force of nature.


• In particular, gravity governs the motion of the moon and artificial satellites around the earth, motion of the planets around the sun and, of course, the motion of the bodies falling to the earth.

2. The Electromagnetic force:

• Electromagnetic force is the force between the charge particles.


• When charges are at rest, the force is given by coulomb's law; attractive for unlike charges and repulsive for like charges, the magnitude of the force obeying the inverse-square law.


• Charges in motion produce magnetic effects and a magnetic field gives arise to a force on a moving charge.


• Electric and magnetic effects are inseparable hence the name electromagnetic force.


• They are long range forces and hence they also don't need any intervening medium.


• The electromagnetic force between two stationary protons for example is 10³⁶ times then the gravitational force between them, for any fixed distance.


• Gravitational force is always attractive in nature while electromagnetic force can be attractive or repulsive.


• Electromagnetic force also depends upon the medium prevailing between two objects.

3. Strong (nuclear) force:

• This force is responsible for binding protons and neutrons in a nucleus.


• It is evident that without some attractive force, a nucleus will be unstable due to the electric repulsion between the protons.


• The strong nuclear force is the strongest of all fundamental forces, about 100 times the electromagnetic force.


• It is charge, independent and acts equally.


• It is a short range force.


• Recent developments have indicated that this force is a Quark-Quark force.


• Neutrons and protons are being made of quarks.

4. Weak force:

• The weak force appears only in certain nuclear processes such as the β decay of a radioactive nucleus.


• In β decay, the nucleus emits an electron and an uncharged particle called neutrino.


• Thus weak force arises due to the interactions of neutrino with other particles.


• The range of weak force is exceedingly small, of the order of 10^-15m.


• This force is responsible for the decay of free neutrons and mesons.

Recent developments indicate that the electromagnetic force and weak nuclear force are two aspects of a unified force known as "electroweak" force.

Fundamental force of
nature

Name	relative strength	Range	Operates among
Gravitational force	10-38	Infinite	All objects in the universe
Weak Force	10-13	Very short, within nuclear size	Elementary particles(neutrino)
Electromagnetic force	10-2	Infinite	Charged particles
Strong nuclear force	1	very short, within nuclear size	Nucleons (neutrons and protons)

Law of conservation of energy

In classical mechanics space and time were considered independent of each other while according to the theory of relativity given by Einstein, space and time are interrelated.

Space is homogeneous and isotropic and hence we have the laws of conservation of linear momentum and angular momentum. Similarly, time is also homogeneous and isotropic. Due to homogeneity of time we have the law of conservation of energy. According to P.A.M. Dirac, one of the great theoretical physicists of 20th century, the law of conservation of charge may be due to the isotropy of time.

This laws are as under:

⇒ Law of conservation of energy:

The amount of total energy in the universe remains constant. The energy can neither be created nor be destroyed; it can just be converted from one form to the another.

⇒ Law of conservation of charge:

During any process taking place in an electrically isolated system, the algebraic sum of the charges always remains constant.

⇒ Law of conservation of linear momentum:

If the resultant external force on a system is zero, the total linear momentum of the system remains constant.

⇒ Law of conservation of angular momentum:

If the resultant external torque acting on a system is zero, the total angular momentum of the system remains constant.

Measurement and system of units

How a unit of a physical quantity should be?

• The measurement of a unit should be definite and unambiguous.


• The unit should be such that its measure does not change.


• The prototype (replica) of a unit should be easily reproducible.


• The replica of a unit should be easily available.

Units of a physical quantities and systems of units:

A limited number of physical quanitities called fundamental quantities or base quantities of which units should be fixed and with the help of them the units of all other quantities can be fixed. The units of fundamental quantities are called as fundamental or base units. The units of all other physical quantities can be expressed as a combination of base units. Such physical quantities are called derived quantities and their units are called derieved units.

The different systems of units are as under:

• British (FPS) system (foot, pound, second system)


• CGS system (centimeter, gram, second system)


• MKS system ( Meter, Kilogram, second system)


• MKSA system ( Meter, kilogram, second, Ampere system)


• SI system

International system of units:

Sr. No.	Physical Quantity	Name of unit	Symbol
1	Length	meter	m
2	Mass	kilogram	Kg
3	Time	second	s
4	Electric current	ampere	A
5	Thermodynamic Temperature	Kelvin	K
6	Luminous intensity	candela	cd
7	Quanitity of matter	mole	mol

Practical Norms for the use of SI system

1. Unit of every physical quantity should be represented according to its symbol.


2. No full stop should be used within or at the end of the symbol for a unit. For example, for kilogram, kg should be written instead of kg. or k.g.


3. Symbols for unit do not take plural form. For example m is used to denote many meters also.


4. The units of physical quantities in numerator and denominator should be written as one ratio only. For example the SI unit of acceleration should be written either as m/s²
   or m s^-2; but not as m/s/s.


5. Full name of a unit, when it is named after a scientist, is not written with a capital letter; but the symbol for that unit has a capital letter. For example, the unit of force should be written as newton but in symbol it is written as N. The symbol for the unit of pressure (viz. pascal) is written in symbol as Pa.

Error's in measurement

When different physical quantities are measured in a laboratory with the help of different apparatus, there would be some in accuracies in the measurement which must be mentioned along with the result.

The inaccuracy in the measurement is called error.

The errors in measurement can be broadly classified as

1. Systematic error


2. Random error

Systematic error:

The errors that tend to be in one direction, either positive or negative. such errors cannot be both, positive and negative simultaneously.

Some of the sources of systematic errors are:

⇒ Instrumental errors:

The errors occuring due to the imperfect design or improper calibration of the instruments.

⇒ Errors due to method of experiment:

For e.g. while measuring the temperature of the human body an incomplete contact of a thermometer with the body causes error in the measurement.

⇒ Personal Error:

Such error arises due to an individual's carelessness in taking observations or due to the fauly method.

Random Error:

The errors which arise due to random and unpredictable fluctuations in experimental conditions are called random errors.

Electric charge

Matter consists of many (more than 100) fundamental particles. Three out of them are most important, namely electron, proton and neutron. There masses are

⇒ m_e = 9.1 x 10^-31
⇒ m_p ≈ m_n = 1.6 x 10^-27

These particles will attract each other due to gravitational forces.

⇒ This gravitational forces acts upon it according to the Newton's universal law of gravitation.

⇒ At the same time there also acts another force of repulsion between them. This force, in addition to the gravitational force is the electric force.

⇒ The fundamental intrinsic property due to which such a force acts is called the electric charge.

⇒ Charges are of two types. Any one of them is considered positive and the other negative.

⇒ Traditionally charge of proton is considered positive and that on an electron negative.

⇒ The force acting between the like charges is repulsive and it is attractive between two unlike charges.

⇒ In any substance electrons are comparatively weakly bounded thus when there is exchange of charge between two bodies, electrons are transferred from one body to another.

⇒ Coulomb is the SI unit of the quantity of charge and is represented by C.

⇒ The Quantity of charge passing in 1 second, through any cross section of a conductor carrying 1 ampere current is called 1 coulomb .

⇒ The magnitude of charge on an electron and on a proton is 1.6 x 10^-19
⇒ Electric charge, like mass is a fundamental property which is difficult to define.

Quantization of Electric charge

⇒ All the experiments carried out so far indicate that the magnitudes of all charge found in nature are in integral multiple of a fundamental charge ( Q = ne)

⇒ This fact is known as the quantization of electric charge.

⇒ This fundamental charge is the charge of an electron which is denoted by e and it is called the fundamental unit of charge.

⇒ The building blocks of all matter, the particle having charge posses charge equal to e.

⇒ For e.g. a positron (positive electron), unlike electron hs positive charge.

⇒ Any atom, on the whole, appears electrically neutral as the number of electrons and protons are normally equal.

⇒ Now it is beleived that the protons and neutrons consist of more fundamental particles called Quarks.

⇒ These quarks are of two types: the quark possessing +2/3 e charge is called an up quark, the one having -1/3 e charge is called a down quark.

⇒ Matter is formed of such quarks and electrons.

⇒ A proton and a neutrons are formed out of a combination of three quarks.

⇒ Other types of quarks are also discovered which are responsible for the formation of other unstable fundamental particles having uncommon properties.

Conservation of electric charge

The law of conservation of electric charge:

Irrespective of any process taking place, the algebraic sum of electric charges in an electrically isolated system always remains constant.

⇒ In an electrically isolated system, a charge can neither enter from out side nor escape from inside.

⇒ In an electrically isolated system, only those processes are possible in which charges of equal magnitude but unlike charges are either produced or destroyed.

Charging by induction

⇒ Consider two identical isolated spheres, one carrying charge Q and the other having no net charge.

⇒ If they are brought in contact and are seperated , the two sphere will have equal amounts of electric charge Q/2 after seperation.

⇒ It can be said that the charge is established on the other sphere which is equal to Q/2.

⇒ There is another method of charging any substance.

⇒ Suppose the net electric charge on the sphere is zero.

⇒ A plastic rod rubbed against fur is brought close to the sphere.

⇒ The free electrons on the sphere , as result of repulsion go to the part of the sphere away from the rod.

⇒ Consequently the part of the sphere closer to the rod becomes positively charged.

⇒ Now when the sphere is connected to the earth through conducting rod, the electrons on the sphere are conducted to the earth.

⇒ Still the sphere retains the positive charge even if the connection with the earth is removed.

⇒ When the plastic rod is moved away from the sphere, the electrons get redistributed on the sphere such that the same positive charge is spread all over the surface of the sphere.

In this way a body can be charged without bringing in physical contact with another charged substance. This phenomenon is called Induction of electric charge.

Coulomb's law

The law is as under:

"The electric force (coulombian force) between two stationary point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them."

⇒ According to the law of electric force between the two point charges q₁ and q₂, seperated by a distance r can be given as under:

F ∝ q₁q₂/r²

∴ F = k q₁q₂/r²

Where,

K = proportionality constant ≈ 9 x 10⁹ Nm²C^-2

⇒ In many formulas of electricity, k is replaced by 1/4Πε₀ to avoid writing a factor of 4Π, when ε₀ is the electrical permitivity of free space.

ε₀ = 8.854185 x 10^-12 ≈ 8.9 x 10^-12 C²N^-1m^-2
Hence, F = 1/4Πε₀ X q₁q₂/r²

Conductors, Insulators and Intrinsic Semiconductors

⇒ The elements in the first three groups of the periodic table like the alkali metals, noble metals etc are the good conductors.

⇒ They posses free electrons and their electrical resistance is quite less.

⇒ Non-metals are almost bad conductor of electricity as they don't possess any free electrons.(except graphite)

⇒ The elements in the fourth group of the periodic table like the Si and Ge have greater electrical resistance than the good conductors but have a lower resistance than the bad conductors.

⇒ Such elements are known as Semiconductors.

⇒ The mechanism of electrical conductivity is different in case of the good conductors and the bad conductors.

⇒ Semi-conductors behaves as bad conductors at zero kelvin temperature in their pure form.

⇒ The resistivity depends on temperature as in good conductors resistivity increase with temperature while in semi conductors resistivity descreases on increasing the temperature upto a certain limit.

⇒ Conductivity of semi conductors is also changed by making radiation incident of suitable frequency.

⇒ The electrical properties of any substance depends on the arrangement of the crystals and their composition of electrons.

9 Great Mental Math Tricks

1. Multiplying by 9, or 99, or 999
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Multiply

ing by 9 is really multiplying by 10-1.

So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81.

Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414.

One more example: 68×9 = 680-68 = 612.

To multiply by 99, you multiply by 100-1.

So, 46×99 = 46x(100-1) = 4600-46 = 4554.

Multiplying by 999 is similar to multiplying by 9 and by 99.

38×999 = 38x(1000-1) = 38000-38 = 37962.
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2. Multiplying by 11
===================
To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.

Let me illustrate:

To multiply 436 by 11 go from right to left.

First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.

Write down 9 to the left of 6.

Then add 4 to 3 to get 7. Write down 7.

Then, write down the leftmost digit, 4.

So, 436×11 = is 4796.

Let’s do another example: 3254×11.

The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.

One more example, this one involving carrying: 4657×11.

Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).

Going from right to left we write down 7.

Then we notice that 5+7=12.

So we write down 2 and carry the 1.

6+5 = 11, plus the 1 we carried = 12.

So, we write down the 2 and carry the 1.

4+6 = 10, plus the 1 we carried = 11.

So, we write down the 1 and carry the 1.

To the leftmost digit, 4, we add the 1 we carried.

So, 4657×11 = 51227 .

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3. Multiplying by 5, 25, or 125
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Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.

12×5 = (12×10)/2 = 120/2 = 60.

Another example: 64×5 = 640/2 = 320.

And, 4286×5 = 42860/2 = 21430.

To multiply by 25 you multiply by 100 (just add two 0’s to the end of the number) then divide by 4, since 100 = 25×4. Note: to divide by 4 your can just divide by 2 twice, since 2×2 = 4.

64×25 = 6400/4 = 3200/2 = 1600.

58×25 = 5800/4 = 2900/2 = 1450.

To multiply by 125, you multipy by 1000 then divide by 8 since 8×125 = 1000. Notice that 8 = 2×2x2. So, to divide by 1000 add three 0’s to the number and divide by 2 three times.

32×125 = 32000/8 = 16000/4 = 8000/2 = 4000.

48×125 = 48000/8 = 24000/4 = 12000/2 = 6000.

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4. Multiplying together two numbers that differ by a small even number
==================================

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This trick only works if you’ve memorized or can quickly calculate the squares of numbers. If you’re able to memorize some squares and use the tricks described later for some kinds of numbers you’ll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.

Let’s say you want to calculate 12×14.

When two numbers differ by two their product is always the square of the number in between them minus 1.

12×14 = (13×13)-1 = 168.

16×18 = (17×17)-1 = 288.

99×101 = (100×100)-1 = 10000-1 = 9999

If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.

11×15 = (13×13)-4 = 169-4 = 165.

13×17 = (15×15)-4 = 225-4 = 221.

If the two numbers differ by 6 then their product is the square of their average minus 9.

12×18 = (15×15)-9 = 216.

17×23 = (20×20)-9 = 391.

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5. Squaring 2-digit numbers that end in 5
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If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.

35×35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3×4 = 12 and that’s the rest of the product. Thus, 35×35 = 1225.

To calculate 65×65, notice that 6×7 = 42 and write down 4225 as the answer.

85×85: Calculate 8×9 = 72 and write down 7225.

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6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10
======================================

============
Let’s say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.

An illustration is in order:

To calculate 42×48: Multiply 4 by 4+1. So, 4×5 = 20. Write down 20.

Multiply together the last digits: 2×8 = 16. Write down 16.

The product of 42 and 48 is thus 2016.

Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.

Another example: 64×66. 6×7 = 42. 4×6 = 24. The product is 4224.

A final example: 86×84. 8×9 = 72. 6×4 = 24. The product is 7224

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7. Squaring other 2-digit numbers
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Let’

s say you want to square 58. Square each digit and write a partial answer. 5×5 = 25. 8×8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you’re squaring together, 5×8=40.

Double this product: 40×2=80, then add a 0 to it, getting 800.

Add 800 to 2564 to get 3364.

This is pretty complicated so let’s do more examples.

32×32. The first part of the answer comes from squaring 3 and 2.

3×3=9. 2×2 = 4. Write down 0904. Notice the extra zeros. It’s important that every square in the partial product have two digits.

Multiply the digits, 2 and 3, together and double the whole thing. 2×3x2 = 12.

Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.

56×56. The partial product comes from 5×5 and 6×6. Write down 2536.

5×6x2 = 60. Add a zero to get 600.

56×56 = 2536+600 = 3136.

One more example: 67×67. Write down 3649 as the partial product.

6×7x2 = 42×2 = 84. Add a zero to get 840.

67×67=3649+840 = 4489.
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8. Multiplying by doubling and halving
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Th

ere are cases when you’re multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.

Let’s say you want to multiply 14 by 16. You can do this:

14×16 = 28×8 = 56×4 = 112×2 = 224.

Another example: 12×15 = 6×30 = 6×3 with a 0 at the end so it’s 180.

48×17 = 24×34 = 12×68 = 6×136 = 3×272 = 816. (Being able to calculate that 3×27 = 81 in your head is very helpful for this problem.)

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9. Multiplying by a power of 2
======================
To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2×2x2×2.

15×16: 15×2 = 30. 30×2 = 60. 60×2 = 120. 120×2 = 240.
23×8: 23×2 = 46. 46×2 = 92. 92×2 = 184.
54×8: 54×2 = 108. 108×2 = 216. 216×2 = 432.

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