Multiplication with Reference: Part 4

Multiplication with Reference: Part 4

Dr Himanshu Shekhar

Kids! Multiplication is such a beautiful mathematical operation that it can be explored in several different ways.

The previous few posts are exploring multiplication of numbers using a reference, which are invariably some power of 10. However, nearness to a power of 10 is one pre-requisite of such calculation. In the current post, reference number is taken as some multiple of power of 10, thus expanding the domain of this mental mathematical technique.

In this post reference can be 20 (2x10), 50 (5x10 or ½ x 100), 60 (6x10), 300 (3x100), 700 (7x100) and so on. The product calculation is again carried out under two heads – left partition and right partition.

Ø  Left partition is decided by multiplication of (i) multiplier of power of 10 and (ii) sum of multipliers minus reference.

Ø  Right partition has number of digits as number of zeros in the base of reference and is given by product of differences of multipliers from reference.

In nutshell, only difference will come in the left partition, where it will be multiplied by a number given by multiple of power of 10. Let us solve examples.

54x56 = 5x6|4x6 = 30|24 = 3024. (Mental mathematics explained in earlier post “Multiplication of Numbers with Compliment Digits”)

54x56 = 5x(54+56-50)|4x6 = 5x60|24 = 300|24 = 300+2|4 = 302|4 = 3024. (Reference is 5x10:, Multiplier 5: Right partition has 1-digit only).

54x56 = ½ x (54+56-50)|4x6 = ½ x60|24 = 30|24 = 3024. (Reference is ½ x100:, Multiplier ½ : Right partition has 2-digit only).

48x63 = 4x(48+63-40)|8x23 = 4x71|184 = 284|184 = 284+18|4 = 302|4 = 3024. (Reference is 4x10:, Multiplier 4: Right partition has 1-digit only).

48x63 = 5x(48+63-50)|-2x13 = 5x61|-26 = 305|-26 = 305|4-30 = 305|34 = 305-3|4 = 302|4 = 3024. (Reference is 5x10:, Multiplier 5: Right partition has 1-digit only: -26 = 34).

48x63 = ½ x(48+63-50)|-2x13 = ½ x61|-26 = 30.5|-26 = 30|50-26 = 30|24 = 3024.  (Reference is ½ x100:, Multiplier ½: Right partition has 2-digit only: 0.5 on left partition = 0.5x100 = 50 on right partition).

48x63 = 6x(48+63-60)|-12x3 = 6x51|-36 = 306|-36 = 306|4-40 = 306|44 = 306-4|4 = 302|4 = 3024. (Reference is 6x10:, Multiplier 6: Right partition has 1-digit only).

312x318 = 31x32|2x8 = 992|16 = 99216. (Mental mathematics explained in earlier post “Multiplication of Numbers with Compliment Digits”)

312x318 = 3x(312+318-300)|12x18 = 3x330|216 = 990+2|16 = 992|16 = 99216. (Reference is 3 x 100:, Multiplier 3 : Right partition has 2-digit only).

798x805 = 8x(798+805-800)|-2x5 = 8x803|-10 = 6424|90-100 = 6424|190 = 6424-1|90 = 6423|90 = 642390. (Reference is 8 x 100: Multiplier 8 : Right partition has 2-digit only: -10 = 190).

798x805 = 4/5 x(798+805-800)|-2x5 = 4/5 x803|-10 = 3212/5 |-10 = 642.4|-10 = 642|400-10 = 642|390 = 642390. (Reference is 4/5 x 1000:, Multiplier 4/5 : Right partition has 3-digit only: 0.4 on left partition = 0.4x1000 = 400 on right partition).

237x245 = ¼ x (237+245-250)|-13x-5 = ¼ x232|065 = 58|065 = 58065. (Reference is ¼ x 1000:, Multiplier ¼ : Right partition has 3-digit only).

758x753 = ¾ x (758+753-750)|8x3 = ¾ x 761 | 024 = 2283/4 | 024 = 570.75 | 024 = 570| 750+024 = 570|774 = 570774. (Reference is ¾  x 1000:, Multiplier ¾ : Right partition has 3-digit only: 0.75 on left partition is 750 on right partition).

Kids can explore themselves with the help of their guardians and teachers to understand the method. Practice makes everything perfect. So, practice and master the technique of multiplication involving references.

Each example can be solved by a number of method and reference. The end result remains same but in between mathematical calculation varies little bit. This mental mathematical techniques, can be helpful in cracking all competitive examinations and also for cracking online examinations, involving numerical aptitude.

Dr Himanshu Shekhar

Multiplication of Numbers with Compliment Digits

Multiplication of Numbers with Compliment Digits

Kids! This post is for finding multiplication of numbers, fulfilling certain criteria.

In mathematics, Ancient India is credited with invention of Zero and that led to development of decimal system. This is because there are only 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to constitute entire mathematics. The topic in this post discusses simple method for multiplication of two-digit numbers, whose unit place digits are 10-compliment of each other and other digit is the same. This means that the last (Unit-place) digits of the multipliers add to 10. The steps are as follows:

• Partition the product into two parts.
• Left part is given by product of tens-place digit and one more than the 10s-place digits
• Right side is given by product of unit place digit and two places are allocated for this.
• If right side partition exceeds 100, carry over to left side is permitted.

Since it is mathematics and not literature, without much writing, some examples are solved directly for those multiplications, where unit place digits of multipliers are 10s compliment and other digits are the same.

Suppose 92x98 is to be obtained. Both numbers have 9 at tens place and sum of unit place digit is (2+8 =) 10. Left side of the product is 9x10 = 90 and right side of the product is 2x8 = 16. So, 92x98 = 9016.

92x98 = 9x10 | 2x8 = 90 | 16 = 9016

83x87 = 8x9 | 3x7 = 72 | 21 = 7221

64x66 = 6x7 | 4x6 = 42 | 24 = 4224

71x79 = 7x8 | 1x9 = 56 | 09 = 5609

52x58 = 5x6 | 2x8 = 30 | 16 = 3016

More examples, like 91x99 (= 9009), 93x97 (= 9021), 94x96 (= 9024), 81x89 (= 7209), 82x88 (= 7216), 84x86 (7224), 72x78 (= 5616), 73x77 (= 5621), 74x76 (= 5624), 61x69 (= 4209), 62x68 (= 4216), 63x67 (= 4221), 51x59 (= 3009), …. can be tried by kids and the power of quick mathematics can be explored.

This type of mathematical ease has many types of corollaries.

TYPE I: The process can be extended to three digit numbers also. 113x117 = 11x12 | 3x7 = 132 | 21 = 13221. 122x128 = 12x13 | 2x8 = 156 | 16 = 15616.

TYPE II: If the last two digits of three digit number add to 100, the same process is repeated with last four digit earmarked for right hand side. 191x109 = 1x2 | 91x9 = 2 | 0819 = 20819, 198x102 = 1x2 | 98x02 = 2 | 0196 = 20196. 189x111 = 1x2 | 89x11 = 2 | 0979 = 20979. It can be extended to 4-digit numbers too. 1191x1109 = 11x12 | 91x9 = 132 | 0819 = 1320819, 1289x1211 = 12x13 | 89x11 = 156 | 0979 = 1560979.

Type III: The same technique can be used for finding square of a number with 5 at unit place. 25x25 = 2x3 | 5x5 = 6 | 25 = 625, 115x115 = 11x12 | 5x5 = 132 | 25 = 13225, 145x145 = 14x15 | 5x5 = 21025.

TYPE IV: The process can be extended to any number of digits, but the criteria of compliment of power of 10 is to be satisfied. This means, last few digits must add to 10, 100, 1000, 10000, 100000, and so on. 12399 x 12301 = 123x124 | 99x01 = 15252 | 0099 = 152520099, 129999x120001 = 12x13 | 9999x0001 = 132 | 00009999 = 13200009999.

Kids! Explore this and intimate concern, if any.

Dr Himanshu Shekhar

Introduction to Numbers

Introduction to Numbers

“Mathematics is the queen of sciences and the theory of numbers is the queen of mathematics”. – Gauss.

Anybody and everybody, who has undergone formal education at any level has faced the frightening effects of draconian subject called “Mathematics”. Although Mathematics has only 10 numerals ranging from zero to nine (0,1,2,3,4,5,6,7,8,9), the types of problems that can trouble any individual are magnanimous. Mathematics deals with properties and interaction of numbers and ten characters enlisted above gave the most popular number system a name “Decimal System”.

Numbers are symbolic representations for counting objects. It brings uniformity and makes various dimensions repeatable, independent of time, place, person or domain. Whether it is cost of objects or number of items or weight of an object or volume or area of a domain, numerals are frequently used to denote quantitative parameter for any dimension. But contrary to the popular believe that numbers in general follow a random sequence, a well recognized pattern evolves, when one deciphers the beauty of numerals. In this first part, introduction to numbers is presented in a systematic way.

Each of the ten symbols or characteristic, representing numbers is called digit. But list of numbers does not end with single digit numbers. Biggest single digit number is 9, beyond which two-digit numbers start. Smallest two digit number is 10 and largest one is 99, which becomes a threes digit number from 100 and so on. The smallest digit of any number of digits contain ‘1’ at its extreme left followed by as many zeros as required to complete number of digits. Largest number of given number of digit contains all ‘9’s. For example largest five digit number is 99999 and smallest five digit number is 10000. A very large number is called infinity and very small number is replaced with zero generally. In addition to 10 symbolic representations of numbers, they can be also expressed on a number line, which varies from minus infinity on extreme left to plus infinity on extreme right with centre named as origin situated at number zero. A number lying on left side is smaller.

Numbers are classified under different heads, with certain basic properties. The smallest such set is natural number (denoted by N). All countable numbers from 1 to plus infinity is included in this set (1,2,3,4,5,6,7, …….). If zero is also added to the list of natural number, it is called whole number (denoted by W). In the domain of whole numbers, if negative integers are also added, Integers (denoted by Z; German Zehlan) are formed. A further superset of number is rational numbers (denoted by Q: Quotient). All the numbers, which can be represented in the fractional form p/q, where q is not equal to zero, are included in this series. Naturally this includes all integers. For a fractional form p/q, ‘p’ is called numerator, while ‘q’ is called denominator. Another way of representing rational numbers is the decimal form of either terminating (0.32, 0.25 …) or recurring (0.33333…, 0.166666….) decimal. The numbers, which are not satisfying the fractional or decimal form requirement of rational numbers, are called irrational numbers (denoted by I). It includes square roots (Ö2, Ö3, Ö15 etc), pi (p), base of Naperian log (e) etc. Combination of both rational and irrational numbers forms a set of real numbers (denoted by R). The biggest set of numbers is complex number (denoted by C), which include imaginary part (i = Ö-1) in the set of real numbers.

Based on properties of various numbers, all numbers divisible by ‘2’, are called even numbers (2,4,6,8…….) otherwise they are called odd numbers (1,3,5,7,9….). A number, having only two factors namely ‘1’ and the number itself, are called prime numbers (2,3,5,7,11,13….), while numbers having more than two factors are called composite numbers (4,6,8,9,10,12…). If sum of factors of a number is equal to number itself, it is called perfect number (6 = 1 + 2 + 3; 28 = 1 + 2 + 4 + 7 + 14 …). A number, which satisfies polynomial algebraic equation with integer coefficients are called algebraic number (Integers, square roots, cube roots, trigonometric functions of an angle etc) else they are named transcendental number (log 2, p, 2^Ö2 etc). Reversing digits of a number is writing digits of the number in opposite turned around way. e.g. 14537 is reverse of 73541. If a number and its reverse are same, they are called Palindromes, like 13631, 161 etc.

Do you know?

1.      ‘1’ is neither a prime number nor a composite number.

2.      ‘2’ is only even prime number.

3.      Zero is a rational number.

4.      ‘p’ and ‘e’ are irrational numbers.

5.      ‘p’ is approximated to 22/7 or 355/113 or 3.1415926…

6.      ‘6’ is the smallest perfect number.

7.      There are infinite rational numbers between two rational numbers.

8.      All real numbers can be represented on a number line.

9.      A rational fraction a/b in lowest terms has a terminating decimal expansion if and only if the integer b has no prime factors other than 2 and 5. For integers, ‘1’ remains in the denominator.

10.  Complex numbers cannot be represented on a number line. They are represented in a plane by “Argand Diagram”.

11.  All transcendental numbers are irrational but all irrational numbers (Ö2) are not transcendental numbers.

12.  Trigonometric functions (Sin20, Cos35, …) are algebraic irrational numbers and not transcendental numbers.

13.  Hilbert number (2^Ö2 ) is transcendental irrational number.