Remainder

In this post we will discuss about the methods by which we can find the remainder of a number when divided by any other number easily. We know methods for most of the numbers. But how they are found are prime importance to us, as they can be useful to us. The methods described here can be used to find remainders of any number when they are divided by any other number. We will first give some rules to find the remainders for some divisors.

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2

If the digit at the unit place is zero or multiple of two which is less than 10, i.e. either they are 0,2,4,6 or 8. The remainder of the number is either 0 or 1.

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3

If the sum of the digits is divisible by 3 then the number is divisible by 3. The remainder is the number which is left when the process is repeated again with the obtained sum.

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4

The number is divisible by 4 if the last two digits is divisible by 4. The remainder is the remainder obtained by the division of the last two digits.

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5

If the digit at the unit place is either 0 or 5 then the number is divisible by 5. The remainder is the remainder when the last digit is divided by 5.

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6

If the number is divisible by 2 and 3 both then the number is divisible by 6. It can also be checked by multiplying every digit by 4 and adding and repeat the process till you get a number which is easily divisible by 6. If the remainder left is zero then the number is divisible by 6 else the remainder is the remainder.

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7

If the digits of the number starting from the unit place is multiplied by the digits 1,3,2,6,4,5 which is repeated for the left digits and all the results are added and the process is repeated till you get a number easily divisible by 7. If the remainder left at the last is zero then the number is divisible by 7 else not. The remainder is the remainder found at last.

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8

If the last three digits is divisible by 8 then the number is divisible by 8 else not. The remainder is the remainder found from the last three digits.

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9

If the sum of the digits is divisible by 9 then the number is divisible by 9. If the sum is large then the method can be applied to this number also. The remainder is the remainder of the sum.

The methods depend on the simplified process to find the remainder at each position and then add them. For example if the number is 768 and it is divided by 3. Then the number can be represented as 7×100 + 6×10 + 8. As a number can be represented as dividend = quotient × divisor + remainder. Here a term containing a number×quotient×divisor is always divisible by divisor so we have to check whether number×remainder is divisible by divisor or not. Here, 10 = 3×3 + 1 and it is present in the dividend as 6×10 = 6(3×3 + 1). 6(3×3) is divisible by 3 and 6×1 is also divisible by three. Similarly 700 and 8 are not divisible by 3. 700 = 7×(3×3×11 + 1) is divisible when 7 is divisible and 8 is divisible when 8 is divisible. Add 7 + 6 + 8 = 21 so the number is divisible as 21 is divisible.

We can follow the similar methods for all the numbers. When the remainders for 10^x where x is an integer gives a pattern then the pattern can be used for making the methods simple.

To have a detailed description visit https://sites.google.com/site/rahulsmaths/idiscover/2013/remainder.

Let us have an example of divisibility by 7.

Divisibility by 7

Method 1:

The number is divisible by 7 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 7.
Remember the sequence 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1, 3, 2, 6, 4, 5, 1,....
Here 1, 3, 2, 6, 4, 5 repeat in the sequence.

0.Reverse the order of the digits	6	7	8	3	5	6	2	4
1.Multiply with the sequence	6×1	7×3	8×2	3×6	5×4	6×5	2×1	4×3
--	6	21	16	18	20	30	2	12
2.Add the numbers	6+21+16+18+20+30+2+12=125
3.Repeat step 0 with the added number	5	2	1	0	0	0	0	0
4.Multiply with the sequence	5×1	2×3	1×2	0	0	0	0	0
--	5	6	2	0	0	0	0	0
5.Add all	5+6+2 = 13
6.Divide the result by 7 find the remainder	mod(13/7)=6

If the remainder is zero (0) then the number is divisible by 7 else the result is remainder.

The remainder is 6.  Repeat the step 0,1 and 2 till you get a number whose remainder you can find easily.

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Method 2:

The number is divisible by 7 if the number follows the following method and the remainder is zero.

Let us find the remainder of 42653876 when divided by 7.
Remember the sequence 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1,....
Here 1, 3, 2, -1, -3, -2 repeat in the sequence.

0.Reverse the order of the digits	6	7	8	3	5	6	2	4
1.Multiply with the sequence	6×1	7×3	8×2	3×-1	5×-3	6×-2	2×1	4×3
--	6	21	16	-3	-15	-12	2	12
2.Add the numbers	6+21+16-3-15-12+2+12=27
3.Repeat step 0,1 and 2 with the result, if number is very large and positive
6.Divide the result by 7 find the remainder	mod(27/7)=6

If the remainder is zero (0) then the number is divisible by 7.

Repeat the step 0, 1 and 2 till you get a number whose remainder you can find easily.

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Remainder

1. In method 1 we can surely say that the result is the remainder of the original number.
2. In method 2 we are not sure for the remainder. If the result is positive then its remainder is the remainder if the result is negative then 7+negative value is the remainder.

Linear equations

Linear equations are those equations which are linear in its variables i.e. the power of the variables are 1. There can be many kinds of linear equations depending upon the number of variables used. a single variable linear equation has one variable, a double variable linear equation has two variables, a triple variable linear equation has three variables and so on. The linear equations with two or more variables is called a multi-variable linear equation. A single variable linear equation looks like this: ax = b where x is the variable. A double variable linear equation looks like this: ax + by = c and a triple variable linear equation looks like this: ax + by + cz = d. The number of variables can extend to any number of variables.

The solution of a linear requires as much equations as there are variables in it. A single variable linear equation requires one equation. a double variable linear equation requires two variables and a three variable linear equation requires three equations. The number of variables is equal to the number of equations. When we find the solution of a set of linear equations then the solutions can be unique or dependent or no solution. The solution of a linear equation can be found with the help of substitution, elimination or with the help of determinants. The solution using determinants is very cumbersome so to simplify the steps we use Gauss algorithms which use matrix to solve such equations. Let us look at the solutions of some linear equations.

Single variable linear equation
A single variable linear equation ax = b has solution x = b/a. When we plot it on a graph then we get a straight line parallel to y-axis. As the value of y is not present in the equation therefore the solution is independent of y.

Double variable linear equation

A double variable linear equation can be solved either by substitution method or elimination method. The solution on a graph is the point of intersection of the lines represented by the two equations.

The equations are 2x − y = 3 and 4x + y = −6

Let us solve the above two equations by substitution method and elimination method.

Substitution method

In substitution method we express one variable in terms of the other and substitute it to get value of one variable. Then substitute it in any equation to get value of other variable.

Step 1:Express any one equation in terms of one variable.
2x−y = 3 ⇒ y = 2x − 3
Step 2:Substitute it in next equation.
4x + y = −6
⇒ 4x + 2x − 3 = −6
⇒ 6x = −3
⇒ x = −(1/2) = −.5
Step 3:Substitute value in the first equation.
y = 2x − 3 = 2(−.5) − 3 = −4
So solution is x = −.5 and y = −4

Elimination method

The elimination method is based on the fact that if one variable is removed from the equation then the reduced equation contains only one variable. After we can find the solution. Substitute one solution in one equation to get the other solution.

Step 1: Eliminate one variable by making coefficient of that variable equal in two equations.

2x − y = 3
4x + y = −6
Add the two equations
6x = −3
x = −(1/2) = −.5
Substitute in anyone
2(−.5) − y = 3
y = − 4

the linear equations of multi variables is found by Gauss elimination method. We will discuss about such methods in some other posts.

Quadratic Equation

A quadratic equation is an expression with highest power in it equal to 2 and equated to zero. The term with the power 2 must not have coefficient equal to zero. If the coefficient is equal to zero then the equation is not quadratic anymore. Usually a quadratic equation has three coefficients but when all the coefficients are divided by the coefficients of x² then the coefficients left are two. The three coefficients of a quadratic equation are a,b and c. When they are put in the equation the equation looks like ax² + bx + c = 0.
A quadratic graph looks like this

The equation of the above graph is y = x² + x − 6

When all the terms of the quadratic equation are made negative then the equation looks like below

The equation of the above graph is y = − x² − x + 6

The solution of a quadratic equation by algebra is visible to all. Let us find the solution of the quadratic equation by calculus and graph. As we can observe from the graph that the graph of the curve is symmetrical about the line x = -b/2a. we will prove this when we arrive at the solution of the quadratic equation. When we use calculus to arrive at the solution of a quadratic equation we make use of derivatives. Let y = ax² + bx + c. When the expression is differentiated for the first time then we get ^dy/_dx = 2ax + b. We have got the rate of change of slope as a linear function. Differentiate it again. we get ^d2x/_dx² = 2a. We have found the rate of change of slope of the slope as constant. This implies that the change in slope of the quadratic expression is linear. As the second derivative is positive so the point where ^dy/_dx = 0 is point of minimum value of y. The point is x = ^−b/_2a. Let us shift the y axis to this point. i.e. X = x + b/2a

The equation becomes y = a(X − ^b/_2a)² + b(X − ^b/_2a) + c
y = aX² − bX + ^b²/_4a + bX − ^b²/_2a + c
y = aX² − ^(b²−4ac)/_4a

The roots for the above equation is when y = 0 i.e.
aX² − ^(b²−4ac)/_4a = 0
aX² = ^(b²−4ac)/_4a
X² = ^(b²−4ac)/_4a²
X = ±√(b²−4ac)/2a

As X = x + b/2a
x + b/2a = ±√(b²−4ac)/2a
x = [− b ±√(b²−4ac)]/2a
The two solutions are x₁ and x₂ where x₁ = [− b + √(b²−4ac)]/2a
x₂ = [− b − √(b²−4ac)]/2a
The above method also tells us that the two roots are equidistant from the axis x = −b/2a. This also gives us an idea that the graph is symmetrical about x = −b/2a because the rate of change of slope is constant. And change is slope is linear.

Life like an equation and astrology

If we consider our universe to be a big computing machine. Then everything in this universe can be guided by equations. Then we can also consider our life to be guided by an equation. And the course of our country or the life of earth as the superposition of all the equations. Which is very similar to Fourier Series except here the building blocks are not sine or cosine but algebraic polynomial of varying degrees.

Now consider a bank. The bank has many customers. The amount of money in the bank depends on its customers. If the money of the customers is at peek then the money with the bank is huge but if the money with the customers is less then the cash pile of the bank is less. As we have considered the life of human beings as equations then the money in the bank is purely guided by money equations of the customers. Here money equation denote the equations which guide the money with a specific human whose equation it is.

Usually it is very hard to access equations of all the human beings present on earth. So we use statistics to arrive at such equations which guide our economy, life , etc. Statistics gives the superposition of all the equations at a point. Now, suppose if there are ten humans on Earth and they constitute the system of earth. Now, if we try to access values of a certain variable like total money present with all of them,then we will have to access ten such points i.e. at ten instants. We may then form 10 equations and such equations can give the values of the constants. But here we don't know whether the amount of money with each human depends on 10 terms with different different coefficient and powers only and they are same in all the equations. But it does not matter as if it depends on more then we will get dependent solutions and they can be useful to find if one or more values are known.

The most important thing which comes in our way is the rate at which the life of a person goes in nature. Like someone acquires height at a very fast pace and some acquires at a very small pace. This thing is found if we look at the life of the person.

Astrology is a kind of solutions to these equations. It details when our life equations with the slopes. It tells when they will be the maximum, when it will be the minimum and when they are smooth. So if one equation of the life of a person is governed by 20 degree polynomial then it may have 19 inflexion points i.e.minimum and maximum. As our universe consists of stars, planets and many other things they guide our life as their life is much smooth compared to our life. They depends on less factors.

At last
Astrology is a simplified science.

Universe a computing machine and human mind

"I say: Our universe is a computing machine."
Our universe is much like a computing machine. But I would say it is a complex computing machine. It obeys all the laws of mathematics in one way or the other. But it can be said that mathematics is the result of this universe. We started our life of mathematics from the visuals of this Universe. WE first studied about numbers then other parts came into existence.

Take the example of two containers. One with 2 l of water and other with 3 l of water. The universe does not even calculate the result but also it checks bounds. Suppose the capacity of first container is 5l and capacity of second container is 4l. Now when we pour water from first container to the second container then the water overflows. But the water is present in this universe which is a bigger container. When we pour water from the second container to the first container then the first container is wholly filled.

There are many examples in physics in which we see instant results of complex calculations. Like when five balls are placed touching each other then the law of momentum is clearly visible. If we move two balls and collide them then two balls are moved from the other end.

This thing gives us an idea that what we see or observe can have a solution. If it happens in this universe it has a solution. Then one thing comes to our mind that can something which a human thinks have a solution. Because as human is also a part of this universe and the thoughts which comes in human mind has some reason. We may think the process of thought is like a number line and one arrives at different points of it following that path then everything which comes in human mind can have a path and if we can trace that path then the solution to that problem exists.

As imaginary number exists and it may be the case that the thoughts can be traced into the domain of imaginary numbers. Or there is any other domain or dimension which gives rise to thoughts in the human mind. And if we can trace that domain then we can read the human mind as we know from which path we must go to reach a human mind.

The questions are many. Let us delve into it and express our thoughts on this blog with the help of mathematics.

Generation of a number system

The method to change from other base to the base 10 is

1. Take i=1, sum = 0 and the rightmost digit as the first digit
2. Repeat each step from 3 to 5 for digits from right to left
3. Calculate (i^th digit)×(base)^i-1 and store in digitvalue.
4. Store sum=sum+digitvalue
5. i=i+1 if (i+1)th digit from right exist else go to step 6
6. The value of the number in base ten is sum.

Now the problem arises when the base ten has no zero symbol. then the above procedure can be written as

1. Take i=1, sum = nil and the rightmost digit as the first digit
2. Repeat each step from 3 to 5 for digits from right to left
3. Calculate (i^th digit)×(base)ⁱ and store in digitvalue.
4. Store sum=sum+digitvalue
5. i=i+1 if (i+1)th digit from right exist else go to step 6
6. The value of the number in base ten is sum/base.

Let us generate a number system which has no zero in it.
Let the base of the number system be three (3). The symbols used are α, β and γ. Where α < β < γ. β = α + α, γ = β + α. The number system looks like this

⇒
⇓	Level 1	α	β	γ
	Level 2	αα	αβ	αγ
		βα	ββ	βγ
		γα	γβ	γγ
	Level 3	ααα	ααβ	ααγ
		αβα	αββ	αβγ
		αγα	αγβ	αγγ
		βαα	βαβ	βαγ
		ββα	βββ	ββγ
		βγα	βγβ	βγγ
		γαα	γαβ	γαγ
		γβα	γββ	γβγ
		γγα	γγβ	γγγ
	Level 4	αααα	αααβ	αααγ
		so.on.

In the above table the numbers increase from left to right and from top to bottom. First the numbers increase from left to right then move to one row below. The numbers in each succeeding level is the result of placing one digit before each number of the preceding level. First α is placed then β is placed then γ is placed. First level consists of 3 numbers, second level consists of 9 numbers and n^th level consists of 3ⁿ numbers. The process to convert the number to base 10 is the same as described above.


You can generate your your number system by this method. The number system may or may not have a zero.

1. Choose the symbols.
2. Define the law of precedence.
3. Describe the step size or relation between symbols.
4. Generate the number system in levels.
5. If the number contains zero. The first series in each level is same as the level before. Remove it.

Let us see how the method works for the above number system.
Step 1: The symbols are α, β and γ.
Step 2: α < β < γ.
Step 3: β = α + α, γ = β + α.
Step 4: Generate as in table.
Step 5: 0 is not present.

Let us convert some numbers from base three to base 10 where α=1.

βα = (β×γ^β + α×γ^α)/γ
βα = (2×3² + 1×3¹)/3
βα = (2×9 + 1×3)/3
βα = 21/3 = 7

γβα = (γ×γ^γ + β×γ^β + α×γ^α)/γ
γβα = (3×3³ + 2×3² + 1×3¹)/3
γβα = (3×27 + 2×9 + 1×3)/3
γβα = 102/3 = 34

Number System

Our ancestors used different kinds of symbols to mean different things. Some of those symbols were used to represent message and some were used to count things. The symbols which were used to count things entered into mathematics after a long phase of change. the symbols which transferred message entered the domain of language used to express what a person speaks. Language also went many phases of change. Here we are talking about numbers. So let us concentrate more on it. The most common number of symbols used in any language for the representation of numbers is 10. Maya civilization used 20 symbols. Actually they were not 20 but the representation was for 20. The use of ten symbols can be attributed due to the numbers of fingers in someone's hand. Many systems existed but the system which was adopted by most of the countries today is Decimal Number System.

With the advent of technology other number systems came into being. Some were used to simplify the task of the representation of older systems. The number of symbols used in a number system is called its base or radix of the number system. Let us look at the most important number systems used today

Decimal Number System

The Decimal Number System uses 10 symbols. Its base is 10. They are 0,1,2,3,4,5,6,7,8 and 9. It is known as Hindu-Arabic System as it was developed in India and modified in Arab. The system initially does not had a zero in it. All the multiples of ten used different symbols. After a lot of time zero was used in representation of the multiples of ten.

Binary Number System

Binary Number System is used in computers. It has two symbols 0 and 1. Its base is 2. Some kind of binary logic was used by Egyptians in the process to find multiplication and division.

Octal Number System

Octal Number System is also used in computers but only as representation and simplification of binary numbers. There are 8 symbols used in Octal Number System. They are 0,1,2,3,4,5,6 and 7. Its base is 8.

Hexadecimal Number System

Hexadecimal Number System has sixteen symbols. The use of hexadecimal number system is similar to octal number system. The symbols used are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F. The letters used are due to the scarcity of symbols. Its base is 16.

Mathematics: Need and areas

"Mathematics quantify our life"
The essential of mathematics is clearly visible from the following paragraph.

A person wakes up. He wants to know what is the elapsed time from the sunrise. He makes a sundial and looks at the relative position. After sometime he starts his journey towards his work place. He reaches there and finds that his one customer waited for him and left for somewhere else. He thought he must have known the time accurately.

Accuracy is the cause of all mathematics. A man can find the amount of bulk investment but how to distribute it in chunks is taught by mathematics. One can find the shortest path from one place to other but what time it will take to reach that place is told by mathematics. One can pass a vehicle moving ahead of it. But how much time it will take to pass it is told by mathematics.There are many such cases in which the estimation is replaced by accuracy with the help of mathematics.

Mathematics is used in all spheres of life. It is used by a boy to the adult and at last by a old person. A shopkeeper uses it keep record of his profits and losses. A carpenter uses it to measure the extent of his furniture. A businessman uses it to increase his profits. A sportsperson uses it for accuracy. A scientist uses it in different fields of research. Such is the extent of mathematics that if you need accuracy then there is no substitute for it.

There are many branches of mathematics such as calculus, algebra, analysis, geometry, trigonometry, probability and statistics, etc.