Number System Tricks & Tips
Number System and basis arithmetic operations
We use decimal number system with base 10. This is the Natural Number System. Any number in this system is created by using ten numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first nine numerals or digits can be used for counting things but 0 is an abstract concept and is used primarily for building numbers larger than 9.
Zero is one of the greatest human inventions and without it, Maths and the civilization as we know it wouldn’t have been there.
Forming a number
Numbers are formed by combining digits of the number system according to a specific mechanism. This mechanism or system is called Place Value Mechanism.
Place Values
- The rightmost digit in a number is identified as the least significant digit, and the leftmost the most significant digit. This is because the place value of the rightmost digit is the least and that of the leftmost digit is the highest.
- The place value of first digit (on the right) is 10^0 which is nothing but 1.
- The place value of the second position is 10 times the place value of the first position. Place value of the third position is 10 times the place value of the second position. Place value of the fourth position is 10 times the place value of the third position and so on. In other words, place value of any position is 10 times the place value of the previous position on its right. Thus place value of position 1 is 10^0 or 1, of second position is 10^1 or 10, of third position is 10^2 or 100, of fourth position is 10^3 or 1000. This is why the rightmost digit is called UNIT’S digit, the second digit from right is called as TEN’S digit, the third digit is called HUNDRED’S digit, the fourth digit is called THOUSAND’S digit and so on. In general, Place value for nth position is, 10^(n−1), where n is an integer (positive, negative or zero, we are not considering now negative values of n which will be used in forming decimal part of decimal numbers).
- Using the place value mechanism (or system) and with the help of 0 and the other nine elementary basic digits 1 to 9, any number, however large (or small), can be formed.
These place values are fixed and are not related to any specific number.
These are independent of any number. This is the basic framework on which the digits of a number are placed to finally form the number.
Place value mechanism is the second most important human invention for Maths, and without our being aware, it supports all the elementary operations on numbers – addition, subtraction, multiplication and division.
Place Values:
Creation of a number
Face Values
A number is formed by multiplying each digit of the number by the Place value of the position of the particular digit and summing up all such products. Let’s understand the structure of the number 217098651.
Each such value of digit contribution is termed as Face Value of the digit.
Creation of a number and Face Values
The digit value contributions of the digits are the face values of the digits. For example, face value of 8 is 8000.
Visualization of Numbers
If you imagine that the numbers start from 0 and go on increasing by 1, the numbers formed in your mind are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. After 9 if you increase the number by 1 it becomes 10 and these numbers 1 to 10 form the first tens numbers. If you still increase the number by 1 serially, you get another nine 11, 12, 13….19 and then 1 more to twenty which form the second tens. The twenties starts from 20 and ends at 29. Increasing 1 we get into the thirties.
This way your mind goes on stepping through the forties, fifties, sixties and so on till it reaches last number of the nineties that is 99. On increasing one more you get the first and lowest three digit number 100. This is the first hundred from 1 to hundred. If you still go on increasing the number by one serially you would reach the second hundred at 200, the third at 300 and the ninth at 900. If you continue to increase the number by another hundred, you would now reach the first thousand at 1000.
The numbers thus go on increasing in steps of 1 then ten then 100 then 1000 and so on. In your mind you form specific positions for the numbers. This is what we call visualization of numbers. For doing mental calculations accurately and fast, this mechanism of visualization of numbers helps a lot. It is generally accepted that at the heart of mathematical capability lies the capability of doing mental maths.
Different Types of Numbers
Broadly numbers can be divided into two categories
- Integers, such as 1, 5 or 86. The set of integers is represented by [ …-1,-2,-3,…0, 1, 2, 3…].
- Decimal numbers, such as 0.345, 67.35.
Decimal numbers have two parts: an integer part on the left of the decimal dot, and the right hand portion of the decimal part starting with the decimal dot and continuing to the right. In 67.35, integer part is 67 and decimal part is 0.35. The number 67.35 is formed by summing up the two parts.
In general one can say,
All numbers are decimal numbers,
with integers having a zero decimal part.
Integers or decimal numbers both have their negative counterparts. For example, positive integers, 1, 2, 3 and so on have their negative counterparts as -1, -2, -3 and so on. Similarly for the positive decimal numbers we have negative decimal numbers. These sets of mirrored numbers, negative integers and negative decimals, together form the negative numbers.
Positive numbers and negative numbers of the forms shown above, together form the set of all possible Real numbers. We won’t consider the opposite of Real, that is, the Imaginary numbers here.
Number line
The complete set of real numbers comprising of positive integers and decimals and negative integers and decimals often is represented visually related to each other as a Number line on which all these numbers are placed in keeping with their values.
This is a very convenient way to visualize the relative values of a set of numbers.
On both sides of the center-point of 00 the numbers go on increasing. On the positive side, the positive integers 1, 2, 3, 4, 5…. go on increasing indefinitely to infinity. Same thing happens on the negative side with negatively increasing negative integers -1, -2, -3, -4. -5……..indefinitely.
The decimal numbers appear between each interval of two integers. 1.5 is just at the middle of the interval between 1 and 2 and 4.5 is at the middle of the interval between 4 and 5. The decimal number 2.25 appears at one-fourth of the interval between 2 and 3 from the position of 2 on the number line. Similarly 3.75 appears at three-fourth of the interval between 3 and 4 from the position of 3.
If you consider carefully you would find that there are,
Infinite number of decimal numbers between any two integers.
We may go on dividing the interval between say 2 and 3 into continuously smaller sub-intervals to create increasing number of decimal numbers between this interval. There is no limit to how small a division you may wish to make.
Just as the integers go on increasing indefinitely to very large to still larger numbers, a single sub-section between two integers may go on decreasing indefinitely.
This complementarity is an inherent property in nature.
Regarding negative numbers these can be regarded as a mirror image of positive numbers with the mirror placed centrally at 0. For each positive integer, there is a negative integer and for each positive decimal there is a negative decimal.
This again is another form of complementarity but in the aspect of nature of numbers and not their absolute values (absolute value of a number is the value irrespective of its sign).
Odd and even integers
Broadly integers can be divided into two groups:
Even numbers
are integers that are divisible by 2, and ending with one of the digits, 2, 4, 6, 8 or 0.
Odd numbers
are integers that are not divisible by 2. For example, 1, 3, 5, 7 and so on are odd integers.
If you add 1 to an even integer, you would always get an odd integer and vice versa.
In general, one can express an even integer by 2q, where q is any non-zero integer and any odd integer by 2q+1, where q again is any non-zero integer.
Special sets of integers
A few special sets of integers are natural numbers and whole numbers.
Natural numbers
are all positive integers excluding 0,
Whole numbers
Natural number set with 0 included,
Sum of first nn natural numbers
The above figure shows the inherent mechanism of summing up the first nn natural numbers where nn is odd and its value is 1111. The actual value of nn is not important. For any odd number of consecutive natural numbers the concept is valid.
Explanation of the elegant concept of summing natural numbers: If the number of consecutive natural numbers is odd, you can always imagine the numbers as lower half of the numbers (excluding the middle number) paired up with the other half of the numbers as shown above. Sum of each pair results in 12 in this case and five such pairs give 5×125×12 or 10 numbers of 6 or 60. The lone 6 in the middle is left unpaired and thus adding it to the already formed sum we get the final sum of first 11 natural numbers as 11 numbers of 6, that is 66.
What is 6 here? This is nothing other than the average of the 11 numbers, that is, the sum of the numbers divided by the number of numbers. It is the number at the middle of the series of 11 numbers here.
The largest number compensates the shortfall of the smallest number from the average 6, and thus when the two are added, we get a result equivalent to two middle numbers. The second highest number similarly compensates the shortfall of the second lowest number from the shortfall with the middle number. Adding the two thus generates two middle numbers.
This goes on for 5 pairs in this case generating equivalent of 10 middle numbers. The lone unpaired middle number when added to this sum completes the process in a nice result of 11 numbers of middle number.
This happens not only with any sequence of odd numbers of natural numbers, but even for any odd number of uniformly increasing series of integers this mechanism works.
For example, the sum of the 11 numbers, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 is (24+4)÷2×11=154(24+4)÷2×11=154.
Without using any formula you can easily form this type of sum by adding the two extreme numbers, dividing by 2 and multiplying the result by the number of numbers.
Simple and quick.
Let us see what happens if the number of natural numbers is even.
In this case of summing up the first 10 natural numbers, as we can see, 5 pairs are formed without any lone middle number left out. The value of each pair in this case is, (10+1)=11(10+1)=11.
For a general nn, this pair sum will be n+1n+1.
How many such pairs will be there?
It would simply be 10÷2=510÷2=5.
So the total in this case is, 5×11=555×11=55. In case of general value of nn, the sum would be,
(n+1)×n2(n+1)×n2.
Most importantly, even if you don’t remember the formulas, remembering the pairing up concepts would be enough for calculating the sum of any number of consecutive natural numbers.
Breaking up an integer into multiple components
Integers can be broken up into multiple components in two ways:
- by expressing it in sum of integers form. For example,
5=4+15=3+223=2×101+3×100
- by expressing it as a product of integers, for example,
6=3×216=2×2×2×2
Sum of integers
The first set of three examples shows an integer in sum of integers form using two methods.
As a sum of face values of the digits of the integer
There is only one such possibility for any integer.
Examples are,
This concept of breaking up any integer is based on the Place value system and is one of the most basic ways to see how a number is formed. This form is frequently used in number problems.
As a sum of all possible combinations of a pair of integers
For 5 we found two ways to express it into sum of a pair of integers. For 6 instead we have,
So we can break up 6 as a sum of a pair of integers in 3 ways. In general, we can break up an integer n as a sum of a pair of integers in n2 ways, where n is even and (n−1)/2 ways where n is odd.
For solving problems in number system, these two methods of expressing an integer as a sum of integers many times result in elegant solutions.
Occasionally for problem solving, we use a third expression of sum of integers.
As a sum of convenient pair of integers
Many a time, we solve a specific problem by breaking up an integer into a large part and a small part suitably. For example, when we are to multiply 28 with 27, we recognize that 28 is very near to 30 and so to avoid time-consuming direct multiplication, we transform the problem to similar but simpler problem :
This almost always is an accurate and fast method of doing multiplication mentally.
Effectively, we have simplified the problem by converting 28 to nearby 30. The number 30 in this case acts as an Anchor number. Any easily computable number such as any multiple of 10 can be used as an Anchor number.
The specific strategy used in this case is a general powerful strategy of problem solving,
Solve a similar and simpler problem
along with the strategy of Problem breakdown or segmentation when we have broken down the number itself as a sum of integers.
Product of integers
In the product of integers form, each of the integers in the product is called a factor of the integer. For example, when we break up say, 60 as,
we call each of the constituent integers in the product 2, 2, 3 and 5 as factors of 60. As 2 is appearing twice, we say 2^2 or 4 is a factor of 60.
Furthermore, as 60 is perfectly divisible by any of its factors, we also say in a complementary way, 60 is a multiple of the integers 3, 4 and 5. This factor-multiple concept is very important in basic mathematics and is used in many situations.
Prime numbers
Thus we arrive at one of the most discussed type of numbers, namely, Prime Numbers.
A prime number cannot be broken up into product of factors, except 1 and itself.
It has only two factors: itself and 1. In a way you can say, all numbers are made up of prime numbers. Truly when we break up a number into its factors, we find out the prime factors only.
A more accurate expression for 60 would be as a product of 2, 2, 3 and 5 rather than 3, 4 and 5.
Examples of prime numbers are,
Fact: It has been proved that there are infinitely many prime numbers, but it is not at all easy to find a new prime number today even with the fastest computer.
Fact: A Prime number though has no factor other than 1 and itself, it can always be expressed as a sum of products.
Recommendation: Do not always think a number as a series of digits only. It may have further mysteries hidden in it.
Non-prime or Composite numbers
All numbers that are not prime are non-prime numbers or composite numbers. These numbers have more than one factor other than itself and 1. For example, 4, 6, 12 are non-prime or composite numbers.
Thus each of these numbers have more than one factor other than itself and so are non-prime or composite numbers. Most of the integers are composite numbers. Prime numbers are far fewer than the composite numbers.
Coprime numbers
Numbers that have no common factor except 1 are called coprime numbers. For example, 12 and 35 have no common factors and are coprime numbers.
Basic Arithmetic Operations
The four most basic mathematical operations are addition, subtraction, multiplication and division. We will see now the salient points of these four most basic math operations.
Addition
The most basic arithmetic operation is addition that has the following properties.
Property 1: Two numbers, variables or quantities are added to each other by the operation of addition.
Property 2: Addition is represented by the “++” symbol.
Property 3: Addition is a binary operation. A binary operation has only two operands on which the operation acts in a certain way and produces a result of operation.
Property 4: It is defined that,
1+1=2
It means, when 1 is added to 1, it (the 1) is increased by 1 resulting in 2. in the same way,
435+54=489
5+3=8
The last means that when 3 is added to 5, the value of 5 increases by 3 resulting in 8. So Addition increases one number by value of the other.
Property 5: The operands in addition must represent quantities of same kind. For example, 2 goats cannot be arithmetically added to 2 cows in easily formed result of 4. In most of our dealings with addition, this “same kind” assumption is implicit.
Property 6: Addition operation is commutative. For example,
5+3=8and also,
3+5=8
Method of addition
Following are the steps in adding two numbers manually with pen and paper.
Step 1: The two numbers are written one above the other with the digits aligned by place of digit, that is, unit’s digit of second number would be written below unit’s digit of first number, ten’s digit of second number would be written below ten’s digit of first number and so on. A line is drawn below the bottom number. The result of addition will be written below this line.
Step 2: The process of addition is started by adding the two unit’s digits first. The result of addition is written below the line and at the unit’s position. That is, addition starts from right.
Step 3: If the result of addition of a particular place becomes larger than 9, (10 or more, it can be maximum 18) the unit’s digit of the result is written on the result line at the place of digits being added and the 1 in the ten’s position of the result is written as CARRY above the digit in the next higher place (one position left) of the number at the top. It signifies that a value of 10 is to be added at the next step of adding the next higher place digits. Adding 10 in the next higher place means adding 1 to the digits in this position. That’s why CARRY is always one when two numbers are added. It cannot be more than 1 when two numbers are added. Only when more than two numbers are added simultaneously, CARRY can be more than 1.
Step 4: In the next step, the two digits in the next higher place are added and if any CARRY from the previous stage is present it also is added. The unit’s digit of the result is written on the result line in the current position and CARRY if any, is written above the digit in the next higher place on the left of the number at the top as before.
Step 5: This process is continued till there are no digit in the bottom or top number left to be added. The result of addition is formed on the result line.
Example: 875923+432568=1308491 is represented by
Subtraction
Subtraction means taking out or reducing. The properties of subtraction are the following.
Property 1: In subtraction, one number or variable or quantity is subtracted from a second number, variable or quantity.
Property 2: Subtraction is represented by “−−” symbol.
Property 3: Subtraction is a binary operation with two operands just like addition.
Property 4: It is defined that,
2−1=1
435−54=381
In both cases, the first number on the left of the subtraction symbol is reduced by the value of the number on the right of the subtraction symbol.
Property 5: Like addition the operands in subtraction must represent quantities of same kind. For example, Cows cannot be subtracted easily from goats by following conventional methods. We will assume this sameness of operands implicitly.
Property 6: Unlike addition, subtraction is not commutative. For example,
5−3=2, but,
3−5≠2.
Property 7: Subtraction has the capability to generate negative numbers. For example, 3−5=−2.
Property 8: Subtraction can be imagined as a special type of addition. that is, addition of a negative second number to the first number. For example,
3−5=3+(−5)=−2
Method of subtraction
For small subtractions we may do it mentally, but for doing a large subtraction we need a formal method.
Let us illustrate the formal method of subtraction by subtracting, 2568 from 5473. The result will be,
5473−2568=2905
Method:
Step 1: First write the number to be subtracted at the bottom of the number to be subtracted from, by aligning its digits place-wise as before. Draw a result line below the number at the bottom. The result will be written below this result line.
Step 2: Subtraction starts from the unit’s place on the right by subtracting the unit’s digit of the bottom number from the unit’s digit of the top number. The result of subtraction is written below the result line at the unit’s position.
Step 3: If the bottom digit is larger than the top digit (from which the bottom digit is being subtracted) just like addition we look at one position left, that is, the ten’s position in this case of subtracting unit’s digits.
In this case of subtraction, we borrow a ten from the top number which is equivalent to borrowing a 1 from the ten’s digit. This BORROW of 1 is written with a minus sign above the ten’s digit of the top number.
Step 4: The borrowed ten is added to the unit’s digit of the top number and now from this sum the unit’s digit of the bottom number is subtracted. In all cases, this will be possible. The result of subtraction is written in the unit’s digit place below the result line.
Step 5: We move one place left to the ten’s place and if there is no BORROW written on this place we subtract the ten’s digit of the bottom number from the ten’s digit of the top number as before.
Step 6: CARRY treatment: Recommended rugged method (that works in case of any combination of values of the two digits involved in subtraction at the current place): If there is a BORROW of -1 at this ten’s digit place where presently subtraction process is going on, we first add 1 to the ten’s digit of the bottom number and then carry out the subtraction of the two ten’s digits, the bottom one from the top.
This is equivalent to subtracting 1 from the ten’s digit of the top number (to take care of the borrowed 10 at the previous step). But we add 1 to the bottom digit to deal with the awkward situation of the ten’s digit of the top number being 0.
Step 7: We repeat this process by moving one place left at every step, till we reach the leftmost digits.
Example:
Multiplication
On multiplying one number by another the result may become large compared to addition and subtraction. Multiplication is at the centrepoint of a major portion of advancement in mathematics. The properties of multiplication are the following.
Property 1: One number, variable or quantity is multiplied with another similar number, variable or quantity to produce the result of multiplication.
Property 2: Multiplication of n1 with n2 is equivalent to adding n1 with itself n2 times.
For example, multiplying 4 by 3 results in 12. This can be written as 4×3=12. In other words we have multiplied 4 by 3. We would have reached the same result if we had added 4 to itself 3 times. So we can say,
Multiplication is nothing but a shorthand form of repeated addition.
Nevertheless, as adding a number to itself many times is a tedious and inefficient way of doing things, multiplication needs to be considered as an independent powerful operation on numbers.
Property 3: Multiplication also is a binary operation with two similar type of operands. As you can guess, we cannot easily multiply 4 cows by 3 goats.
Property 4: Multiplication is commutative. For example,
4×3=3×4=12.
Property 5: For small multiplications, during early schools, number table is to be learned by heart by the child. If a child has memorized the number table well, she need not take help of any other method or calculating aid to arrive at the result of such small multiplications; this capability effectively increases the speed and accuracy of mental math that is very important for proficiency in math.
In later life for success in any competitive selection test, this ability of high speed mental calculation of especially small multiplications turns out to be invaluable.
Number table can be memorized starting from 2×2=4 to 20×20=400, but this creates a large memory load.
Memorizing a much smaller number table is recommended.
Just like addition and subtraction we need a formal method of multiplying one number with another.
Method of multiplication
Step 1: Preparation: Just like addition and subtraction, we first write the two numbers one below the other with their unit’s digits and other digits aligned place-wise. But in this case we write the larger number at the top and the smaller at the bottom for ease of operation. The result line below the two numbers in this case is an intermediate result line, not the final one.
Objective is to multiply the top number with each digit of the bottom number digit by digit starting from its unit’s digit, write the result of this single digit multiplication in one line below the result of multiplication by the previous digit, but shifted leftward by one place compared to the previous result. This is equivalent to 10 times the actual multiplication result for the unit’s digit (say). Shifting the result by one place left effectively represents this 10 times concept. Finally the results of multiplication of the top number by all the digits of the bottom number will be added up.
This method uses place value mechanism beautifully.
Step 2: At the first step, the top number is multiplied with the unit’s digit of the bottom number aligned place-wise and the result is written below the result line.
Step 2.1: To multiply the larger number with the unit’s digit of the smaller number, first the unit’s digit of the larger number is multiplied with the unit’s digit of the smaller number.
Step 2.2: The unit’s digit of the result is written at the unit’s position and the CARRY if any is written at the top one place left.
Step 2.3: Next the Ten’s digit of the larger number is multiplied by the unit’s digit of the smaller number and CARRY if any from the previous step is added. The unit’s digit of the result is written now at the ten’s place of the result line and CARRY if any is written at the top one place left.
Step 2.4: In this manner all the digits of the larger number are multiplied with the unit’s digit of the smaller number.
Step 3: At the next step, The larger number again is multiplied by the ten’s digit of the smaller number. But the result in this case is written below the first result row with one place shifted to left and appending a 0 on the right.
This represents multiplying the the top number by ten times the current digit in respect of the previous digit multiplication, which is actually the case as per the place value of the current digit.
Step 4: The result of multiplication of the hundred’s digit with the larger number is similarly written below the previous result row with shift of one more place left and appending with two zeros on the right to take care of the place value of 100 of the multiplying digit.
Step 5: In this way the larger number is multiplied by each digit of the smaller number starting from the unit’s digit, moving to left step by step till all the digits of the smaller number have been used up.
Step 6: In the last step, all the results of multiplication are added together to form the final result which is written below the final result line drawn below the last result of multiplication.
Example: 473×256=121088
Division
Following are the properties of the division operation.
Property 1: Just as subtraction must exist as a complementary operation to the most basic mathematical operation of addition, division plays the similar complementary role to multiplication. While multiplication increases a quantity, a division operation decreases a quantity.
Property 2: Naming and relationship of entities involved:
A number n1n1 is divided by a second number n2n2 to produce two things: the quotient qq, and the remainder rr.
The number being being divided is called Dividend and the number that divides the Dividend is called the Divisor.
In general, a division is represented by the following relation between the four entities involved, where n1n1 is the dividend, n2n2 is the divisor, qq is the quotient (the result of division) and rr is the remainder of the whole operation of division.
n1=q×n2+rn1=q×n2+r.
Property 3: Factors: If remainder r is zero, we say that the number n2n2 divides the number n1n1 fully.
This also means that the number n2n2 is a factor of the larger number n1n1 expressed as,
n1=q×n2n1=q×n2.
For example, if 13 is divided by 3 we get remainder as 1, but if 12 is divided by 3, it divides fully and the remainder is zero.
13=4×3+113=4×3+1, where remainder of dividing 13 by 3 is 1, but,
12=4×312=4×3, where remainder of dividing 12 by 3 is 0.
In the second case, 3 is a factor of 12. We express this as,
12÷3=412÷3=4.
But we can see, 4 is also a factor of 12 expressed similarly as,
12÷4=312÷4=3
Effectively as we can see, 12 is a product of two factors 3 and 4. The process of breaking up a number in terms of a product of its factors is called Factorization. This we will learn later.
Property 4: In division of two numbers, the dividend may be less than the divisor. In this case, the result will be a decimal number less than 1 (but greater than 0). For example,
dividend < divisor:dividend < divisor: 4÷5=0.84÷5=0.8
This is the case where the division operation generates a new type of number, that is, a decimal number.
Property 5: If the divisor is less than the dividend, the result will be more than 1. In this case, two things may happen:
- The divisor may fully divide the dividend. Then result will be an integer generally more than 1 (if divisor is equal to dividend, result will be 1). Example, dividend ≥≥ divisor, divisor fully divides dividend: 264÷11=24264÷11=24, 11÷11=111÷11=1.
- If the dividend is larger than the divisor but the divisor does not fully divide the dividend, the result will be a decimal number having an integer part ≥1≥1 and also a decimal part. Example, dividend >> divisor, divisor does not fully divides dividend: 5÷4=1.255÷4=1.25, 274÷3=91.33333…274÷3=91.33333…. The first is called a terminating decimal and the second is a non-terminating decimal where the 3’s extend indefinitely. But a specialty of this decimal is that the 3 repeats. Thus it is called a repeating and non-terminating decimal. We will deal with decimal numbers separately.
Property 6: Constraint: In any division, the divisor cannot be 0. If it is so the result becomes undefined.
Property 7: A division is equivalent to repeated subtraction. For example, we get 4 by dividing 12 by 3. If we subtract 3 from 12 repeatedly 4 times, 12 is completely exhausted to 0. This is equivalent to dividing 12 into 4 parts each of value 3. Otherwise we know, 4 numbers of 3’s make 12. This shows the complementarity of division with multiplication.
4×3=124×3=12,
Or, 12÷3=412÷3=4,
Or, 12÷4=312÷4=3.
Property 8: Just as subtraction generated a new class of negative numbers, in the same way, division generates two new classes of numbers: fractions and decimal numbers with decimal part less than 1 but greater than 0.
Fractions are only a special representation of numbers, but not really a new class of numberswhereas decimals are an important new class of numbers. Like decimals, we will deal with fractions separately.
We would use the fraction representation now to show the complementarity of division with multiplication more clearly.
123=4123=4,
Or, 124=3124=3,
Or, 12=4×312=4×3.
Property 9: In a fraction, the number above the dividing line is called numerator and the number below is called denominator. In these cases the numerator is the dividend and the denominator is the divisor.
In the third expression of multiplication, you can bring any of 3 or 4 below 12 leaving the other on the right of the == symbol. We would then get the two divisions as shown in the first two equations.
Property 10: The operation of division is not commutative. This property is similar as in subtraction.
123≠312.123≠312.
Method of Division
All the other three operations of addition, subtraction and multiplication start from the rightmost digit or the unit’s digit and proceed leftwards. But, for efficiency, division starts from the left, that is, from the most significant digit and proceeds rightwards.
Aside: It is possible to carry out a division operation from left to right but the method is extremely cumbersome and generally unusable. The long division method of the following steps is the most convenient method of carrying out a division using pen and paper.
Step 1: From the left a number of digits are selected so that the number thus formed is just more than the divisor, so that when the divisor divides this number, a single digit quotient is formed. This sub-step involves a bit of trial and error.
Step 1.1: Now this part number is divided by the divisor, the single digit quotient is written as the first left digit of the final result quotient and the result of multiplication of this quotient digit with the divisor is written below the part of number being divided with unit’s digit alignment (with the part dividend).
Step 1.2: This result is then subtracted from the part number used as dividend above it and the result of subtraction is written below the subtraction result line as remainder for this particular division operation. Thus the quotient and the remainder is formed out of the first division.
By the nature of the process, this remainder will be either 0 or a number less than the divisor.
Step 1.3: At the next step, the next digit on the right of the part number selected as dividend in the previous stage is brought down from the original dividend and appended to the right of the remainder.
Step 2: Now the divisor will divide this new dividend formed and corresponding quotient digit is appended to the quotient formed till the previous step and the remainder is written as before below the multiplication result.
Step 3: This way the division progresses digit by digit towards the right of the dividend till the last digit is used up. At this stage we get the final quotient and the final remainder.
This method is called the long division method. Any large division may be carried out by following this method carefully step by step. The following shows how this method is carried out in practice.
Example:
Here divisor is 18, dividend is 1217562, quotient is 67642 and remainder is 6. In five steps of division the whole division has been completed by following the systematic long division method.
Let us now actually solve a few problems on numbers and number system. This part will have three sections. Firstly we will work out a few examples to give you an idea of basic number problems. The second section will be the exercise to be done by you and the third will be the answers to exercise sums.
Worked out sums on numbers and number system
Problem example 1. What is the least number that must be added to 73243 to make the sum divisible by 365?
What should be the strategy of solving this problem? What is really the second target, with first being the least number with the given condition?
First conclusion we can draw is: the sum after addition of the target least number must be a multiple of 365 with no remainder.
Second conclusion: after adding the desired least number to 73243, this multiple must be the multiple just crossing 73243 to become the multiple of 365 larger than 73243 but nearest to it.
This second conclusion is in fact the key idea to solve the problem. We can find the multiple of 365 just lesser than the given number by dividing 73243 by 365. But on second thoughts do we really want this multiple?
Third conclusion: What we really want is the shortfall of 73243 from the multiple of 365 just greater than it. That will give our answer.
Fourth conclusion and action: to get the shortfall, we need the remainder of dividing 73243 by 365 and find how much lesser is the remainder from 365. If we add this number to 73243 we would reach the multiple of 365 just greater than 73243.
On examining 73243 and 365 mentally, we find twice 365 is 730 giving 243 as the final remainder. Again mentally we subtract 243 from 365 getting the answer as 122.
73243+122=73365=201×36573243+122=73365=201×365.
To check, multiply 365 by 200 giving 73000 and add 365 once more to it giving, 73365. Subtract 122 from 73365 and you will get back the given number 73243.
To visualize this situation it is always good to place the numbers and shortfall on a line mentally as below.
Problem example 2. What is the least number of 4 digits that is a multiple of 21?
From our experience gained by solving the first problem, we can make our first conclusion as: the target number must be the multiple of 21 just greater than 1000 where 1000 is the least number of 4 digits.
The method then should be same as in the first problem with given number as 1000 instead of 73243 and factor number as 21 instead of 365.
We try always to solve any such problem mentally as far as possible and so instead of dividing 1000 by 21, getting the remainder, subtracting the remainder from 21 and adding the result of subtraction to 1000 to get the target number, by examination we multiply 21 by 50 to get 1050 and subtract twice 21, that is, 42 to get the answer as 1008. This number is 48 times 21.
Problem example 3. The sum of the ages of Ram and Hari is 96 years. 20 years ago Ram was 3 times as old as Hari. What are their ages?
Solving this type of problems using Algebra is dead easy, but we are not allowed to use Algebra at all here because we are not supposed to know Algebra yet. Instead we would use our mathematical reasoning to solve this problem. If you make a habit to solve number problems using mathematical logic only, you would get an inner view of how the problem gets actually solved. Using artificial aids such as Algebra hides the inner mechanism of problem solving and weakens your problem understanding and solving abilities.
As number problems may be quite tough, if you make a habit of using mathematical reasoning, when you face a tough problem chances would be better for you to see clearly the path to the solution.
Let us face the present problem.
We decide the crucial given information to be the second part, that is, Ram was 3 times as old as Hari. But that was 20 years ago by which period sum of their ages must been reduced by 40 years to 56 years (from 96 years present sum).
It means 56 years consists of 4 parts, one part is contributed by Hari and three parts by Ram. So one part is 14. This was then Hari’s age 20 years ago and three parts must be 42 which was Ram’s age 20 years ago. Presently their ages are then, 62 years and 34 years.
We would go over to the exercises now. Some of these you may not find easy, but be sure that all of these sums can be solved using the concepts we have gone through here.
You will find the answers at the end of the exercise problems.
Exercise problems on Numbers and Number System
Problem exercise 1. Find the sum of first 9 natural numbers.
Problem exercise 2. Find the sum of first 999 natural numbers.
Problem exercise 3. A farmer has some hens and some rabbits. If the total number of the feet of the hens and the rabbits is 160 and their total number of heads is 50, how many hens does the farmer have?
Problem exercise 4. If the numbers 1, 2, 3, 4, 5, …….998, 999, 1000 are multiplied together, how many 0s on the right the product will have?
Problem exercise 5. In a two digit number, the unit’s digit exceeds ten’s digit by 2 and the product of the sum of the digits and the number is 144, then what is the number?
Problem exercise 6. The sum of 37 consecutive positive integers is 851. Find the largest.
Problem exercise 7. Find the unit’s digit of 459×459+77×77+2785×2789459×459+77×77+2785×2789.
Problem exercise 8. In three three digit numbers each of which when divided by 5 leaves a remainder of 3. The difference between the largest and smallest is twice the difference the first two. If the sum of the three numbers is 429, find the second largest among the three.
Problem exercise 9. Find the sum of all the three digit numbers each of which when divided by 5 leaves a remainder of 3.
Problem exercise 10. The ten’s digit of a two digit number exceeds the unit’s digit by 5. The second number formed by reversing the digits is less than the first number by five times the sum of the digits of the number. Find the sum of the digits.
GENERAL RULE FOR SOLVING PROBLEMS IN ARITHMETIC
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