User:Kwenge Erasmus

The uniqueness of the number 9

From time immemorial, numbers have been playing a major role in enabling man to solve problems which he encounters and be able to understand the world he lives in and beyond. For ages, the properties of numbers have fascinated both professionals and laypersons. Number theory has long been recognized as perhaps the only area of mathematics in which amateurs have made significant contributions and their interests in number lie far beyond mere utilitarian applications (Balmond, 2000). Early philosophers and mathematicians such as Pythagoras were able to explain the chaos of human existence with the help of numbers. Aristotle and Pythagoras believed that all things are numbers, and for Aristotle all things could be numerically expressed. The weight of 4 anvils which were proportional to 6, 8, 9 and 12 made Pythagoras believe that all creation existed with similar numerical interrelatedness (Into the Unknown- Readers Digest, n.d). Hence his conclusion that elements of numbers are the elements of all things, and the whole heaven was a musical scale and a number.

According to Griffiths (1972), to find if a number is divisible by 9, one has to add all the digits in the number so that the answer or the sum has only one digit. If the sum of the digits is divisible by 9, then the number is divisible by 9.

Example 1.

Show that 423 is divisible by 9. To show that 423 is divisible by 9, add all the digits to get one digit. 4 + 2 + 3 = 9. Since the sum of the digits is 9, then 423 is divisible by 9

Example 2.

Show that 92 142 is divisible by 9. To show that 92 142 is divisible by 9, add all the digits to get one digit. 9 + 2 + 1 + 4 + 2 = 18 and 1 + 8 = 9. Since the sum is 9, then 92 142 is divisible by 9.

For Balmond (2012), the process of “casting out nines”, where a number modulo 9 is determined by adding its digits; e.g., 34 mod 9 = 7 while 345 mod 9 = 12 mod 9 = 3. Professional mathematicians might be inclined to drop the subject here. However, to do so would be to miss the many beautiful patterns and discoveries about numbers that are to be found by following the path of the number 9 wherever it leads. The key lies in a circular pattern based on multiplying the series 123456789 successively by 1, 2, 3, 4, 5, 6, 7, 8, 9 modulo 9. Herein lies the “Sigma Code”, sigma referring to the process of summing the digits of a number in order to write it as mod 9. In the sigma code, numbers are viewed either statically on a wheel with nine orbits on which the axis of symmetry and the outer orbit are all 9s, or dynamically as outward and inward spirals in which the number 9 is the fixed point of the wind (Balmond, 2012).

Writing about 9 makes an entertaining read as it is an excellent way to interest young minds to discovery of the wonders of numbers and invite non-mathematicians into the joy of making their own discoveries.

With Harold (1982), when the digits of any number are reversed, the difference will be the multiple of 9, for example; 175 if reversed it will be 571. Subtract 175 from 571, the difference is 396. If 483 reversed, it becomes 384. Subtract 384 from 483, the difference is 99.

According to Foster, Cummins and Yunker (1981), if                                is changed to a decimal, there will be the remainder of 9 which repeats and the same quotient digit, 3, will keep repeating. The decimal form for   is 0.333……. this shows that if the repeating decimal has to be expressed in fraction form the denominator will be a multiple of 9.

Change 0  to a fraction.

Let x = 0.

Multiply both sides by 1000

1000x = 345.

x =     0. subtract the original equation

999x  = 345       divide both sides by 999

x =

x =

x  =

The fraction form of 0. is. The denominator of this fraction is a multiple of 9. This again shows that there is a special relationship between repeating decimals and the number 9.

The Kwenge’s formula for π which is:  π = sinθ(1.8 x 10n+2) where θ = 10-n and n the number of decimal places of  required can also be written as    =   180 sinθ(10k) where θ =   and k is the number of decimal places of  required (Kwenge, Mwewa & Mulenga, 2015) is.From the formula, it can be seen that 9 is a factor of π because the factors of π are 9 and sinθ(0.2 x 10n+2) which again makes 9 to be unique. When you multiply any number by 9, and add the product of the two  numbers to one digit number the result or the sum will always be 9 (( http://www.biblestudy.org/bibleref/meaning-of-numbers-in-bible/9.html For example, 6 x 9 = 54, reduce 54 to a single digit by adding them together: 5 + 4 = 9. Similarly, 8 x 9 = 72, and 7 + 2 = 9. Or 23 x 9 = 207, 2 + 0 + 7 = 9, and so forth. There is nothing coincidental about this peculiarity. Any number, no matter how large, multiplied by 9 reduces to 9. According to Hans ( https://www.numerology.com/numerology-numbers/9 ) any number increased by a factor of 9 loses its own identity and instead takes on the characteristics of the 9. There is no number with such characteristic behaviour. A different, but no less distinctive and revealing, attribute of the 9 is that when you add (as opposed to multiply) it to any other number, then reduce that number to a single digit, it always comes back to itself, as if nothing was added at all. For example, 5 + 9 = 14, 1 + 4 = 5. Or 7 + 9 = 16, 1 + 6 = 7. If 24 is added to 9 the answer is 33 and 33 added to a single digit the answer will be 6. In the bible, the number 9 symbolises divine completeness or conveys the meaning of finality. Christ died at the 9th hour of the day, or 3 p.m., to make the way of salvation open to everyone ( http://www.biblestudy.org/bibleref/meaning-of-numbers-in-bible/9.htmlNine also represents the fruits of God's Holy Spirit, which are Faithfulness, Gentleness, Goodness, Joy, Kindness, Long suffering, Love, Peace and Self-control (Galatians 5:22 - 23). Hosea, who was Israel's last king before the kingdom fell to the Assyrians in 723 B.C., reigned for just 9 years (732 to 723 B.C.). The first battle mentioned in God's word is between a confederation of 4 kings against another which has 5 kings for a total of nine (Genesis 14:1 - 2). It was at the 9th hour of the day that a Roman Centurion named Cornelius was told, in a vision, to contact the apostle Peter. Cornelius was eventually baptized and received God's spirit, becoming the first recorded Gentile convert to Christianity (Acts 10).

The energies and attributes of the number 9 come into play throughout the month of September, making it a month of conclusions and endings, opportunities to see things from a clear perspective, integration and the ‘three worlds’ which are the  physical (what we see), intellectual and spiritual ( http://numerology-thenumbersandtheirmeanings.blogspot.com/2011/05/number-9.html ). It is the vibration of the number 9 that gives us the ability to understand our own personal truths and live with integrity. September is a month where you may consider all you’ve achieved (or not achieved) in the year and then make appropriate plans to either keep up the positive flow, or change the situation/s if they’ve not been running as you’d like them to. September is the month for enforcing much needed changes as you approach the last quarter of the year.

The constructivism theory of learning which according to social constructivist scholars view learning as an active process where learners learn to discover principles, concepts and facts for themselves, hence the importance of encouraging guesswork and intuitive thinking in learners (Brown T., McNamara O., Olwen H., & Jones L.,1989). In fact, for the social constructivist, reality is not something that we can discover because it does not pre-exist prior to our social invention of it. Kukla (2000) argues that reality is constructed by our own activities and that people, together as members of a society, invent the properties of the world. Other constructivist scholars agree with this and emphasise that individuals make meanings through the interactions with each other and with the environment they live in. Knowledge is thus a product of humans and is socially and culturally constructed (Ernest, 1991). Learning is a social process and it is not a process that only takes place inside our minds, nor is it a passive development of our behaviours that is shaped by external forces. Meaningful learning occurs when individuals are engaged in social activities. In fact, constructivism is a theory that describes how learning takes place. The theory of constructivism suggests that learners construct knowledge out of their experiences. It is experience that makes the learner to be creative in learning situations. Creativity is about generating new and useful ideas and rules. Engagement in creativity would make learners be involved in producing something new and useful with respect to the previous knowledge, according to Kwenge (2015). A constructivist view of knowledge implies that knowledge is continuously created and reconstructed so that there can be no template for constructivist teaching (Peterson & Knapp, 1993). One learns mathematics by building on the mathematics that one has previously learned. Constructivism is a learning theory of knowledge which states that humans generate knowledge and meaning from an interaction between their experiences and their ideas (Thakur, 2014). Formalization of the theory of constructivism is generally attributed to Jean Piaget, who articulated mechanisms by which knowledge is internalized by learners. He suggested that through processes of accommodation and assimilation, individuals construct new knowledge from their experiences. In order to understand and appreciate mathematics since it is full of abstract; the teacher must have sound content subject matter knowledge and pedagogical knowledge (Shulman, 1987) which includes understanding of numbers such as 9 so that he can be creative and be able to facilitate the teaching and learning of mathematics with joy. Social constructivism or socio-culturalism encourages the learner to arrive at his or her version of the truth, influenced by his or her background, culture or embedded worldview. Historical developments and symbol systems, such as language, logic, and mathematical systems, are inherited by the learner as a member of a particular culture and these are learned throughout the learner's life. This also stresses the importance of the nature of the learner's social interaction with knowledgeable members of the society. Without the social interaction with other more knowledgeable people, it is impossible to acquire social meaning of important symbol systems and learn how to utilise them. Young children develop their thinking abilities by interacting with other children, adults and the physical world. From the social constructivist viewpoint, it is thus important to take into account the background and culture of the learner throughout the learning process, as this background also helps to shape the knowledge and truth that the learner creates, discovers and attains in the learning process (Wertsch, 1997). Furthermore, it is argued that the responsibility of learning should reside increasingly with the learner (Glasersfeld, 1989). Social constructivism thus emphasises the importance of the learner being actively involved in the learning process, unlike previous educational viewpoints where the responsibility rested with the instructor to teach and where the learner played a passive, receptive role. For Glasersfeld (1989), learners should construct their own understanding and must not simply mirror and reflect what they read. Learners look for meaning and try to find regularity and order in the events of the world even in the absence of full or complete information.

Addition and the number 9.

When 2 numbers are being added, first step is to bring together the 2 numbers being added and then subtract the multiple of 9 from the number that has been created by combining the 2 given numbers. To find this multiple, the following formula should be used; let the multiple of 9 be x so that x = 10ka – a.

Formula for addition of any 2 number;

If a and b are whole numbers, then a + b = ab – (10ka – a) where k is the number of digits of b and ab ≠ a x b but a and b brought together in the order they are written.

For example, find the sum of each of the following using Kwenge’s method.

9 + 200

4 + 34

65 + 23

Formula for addition of 2 numbers is; a + b = ab – (10ka – a).

Since b has 3 digits, then k = 3.

9 + 200 = 9 200 – (9 x 103 – 9)

= 9 200 – (9 000 – 9)

= 9 200 – 8 991

= 209

Since b has 2 digits, then k = 2

4 + 34 = 434 – (4 x 102 – 4)

= 434 – (400 – 4)

= 434 – 396

= 38

Since b has 2 digits, then k = 2

65 + 23 = 6523 – (65 x 102 – 65)

= 6523 – (6 500 – 65)

= 6523 – 6435

= 88

General formula for addition of any two numbers;

If a and b are whole numbers, then a + b = ab – (10ka – a), where k is determined by the number of digits of b and ab ≠ a x b but a and b combined or brought together in the order they are written.

Find the following using Kwenge’s method of combination;

5 + 3

3 + 123

45 + 1000

  Method 1                                                               Method 2

Since b has 1 digit, then k = 1, so     (i) Alternative formula;  a + b =  ab – na where n is

a + b = ab – (10ka – a)                     determined by the number of digits of b.

5+ 3 = 53 – (5x10 – 5)                          Since b has 1 digit then n = 9.

= 53 – (50 – 5)                                  Hence 5 + 3 = 53 – 9 x 5

= 53 – 45                                                              = 53 – 45

= 8                                                                        = 8

Since b has 3 digits, then k = 3, so         (ii)  Since 123 has 3 digits then n = 999

a + b =  ab – (10ka – a)                            Hence 3 + 123 = 3 123 – 999 x 3

3 + 123 = 3123 – (103 x 3 – 3)                                              = 3 123 – 2997

= 3123 – (3000 – 3 )                                                =  126

= 3123 – 2997

= 126

Since b has 4 digits, then k = 4, so               (iii) Since 1000 has 4 digits then n = 9 999

a + b = ab – (10ka – a)                                    Hence 45 + 1000 = 451 000 – 9 999 x 45

45 + 1 000 = 451 000 – (104 x 45 – 45)                                        = 451 000 – 449 955

= 451 000 – (450 000 – 45)                                        = 1 045

= 451 000 – 449 955

= 1 045

Kwenge’s method shows how unique is the number 9 and the method can help the learner to develop critical and analytical thinking when dealing with problems. This method can help learners to discover the uniqueness and the powers that each number has.

Subtraction and the number                                        Ordinary method Kwenge’s method

(i)                6 – 2 = 4                                                                 (i)    6 – 2 = 62 – (6 x 9 + 2 x 2)

= 62 – (54 + 4)

= 62 – 58

= 4

(ii)              8 – 5 = 3                                                               (ii)   8 – 5 = 85 – (8 x 9 + 5 x 2)

= 85 – (72 + 10)

= 85 – 82

= 3

(iii)             17 – 6 = 11                                                          (iii) 17 – 6 = 176 – (17 x 9 + 6 x 2)

= 176 – (153 + 12)

= 176 – 165

= 11

(iv)            78 – 23 = 55                                                        (iv) 78 – 23 = 7823 – (78 x 99 + 23 x 2)

= 7823 – (7722 + 46)

= 7823 – 7768

= 55

To find the difference between the two given numbers using Kwenge’s method. First combine the two numbers and then subtract the product of 9 and the first number and the product of the second number and two.

For example, a – b = ab – (9a + 2b) if b is less than 10

a – b = ab – (99a + 2b) if b is more than 10 but less than 100

a – b = ab – (999a + 2b) if b is more than 100 but less than 1000.

General formula for the difference between two numbers using Kwenge’s method:

a – b = ab – (na + 2b) where n is number of 9s determined by the number of digits of b and ab is not equal to a x b but a and b brought together.

if b has 1 digit, then n = 9

if b has 2 digits, then n = 99

if b has 3 digits, then n = 999

Find 135 – 35 using Kwenge’s method.

The formula is : a – b = ab – (na + 2b).

In this case n = 99 because b has 2 digits and ab = 13535.

135 – 35 = 135355 – (135 x 99 + 35 x 2)

= 13535 – (13365 + 70)

= 13535 – 13435

= 100

Division of any number by 9

Let us look at numbers between 1 and 8 inclusive and divide them by 9.

= 0.111111111111………….

= 0.222222222222……………

= 0.3333333333333………………

………………………………………..

…………………………………………

………………………………………….

= 0.8888888888888888…………………..

If n is a whole number which is less than 9, then  = 0.n n n n n n n n n…………

Division by 99 and numbers between 10 and 98.

= 0.212121212121…………………….

= 0.2222222222222……………………

= 0.232323232323…………………..

………………………………………………

………………………………………………

………………………………………………

= 0.98989898989898……………………….

If ab is a whole number which is more than 10 but less than 99, then  = 0. ab ab ab ab…….

Division by 9 and a two digit number.

= 1.1111111111111111……………………………….

= 1.22222222222222…………………………………

= 1.3333333333333333………………………………

= 1.444444444444444444……………………………

= 2.55555555555555555……………………………..

= 5.222222222222222222…………………………….

= 8.6666666666666666………………………………..

If we look at (i) – (v) the sum of the digits of the numerator is less than 9.

Sum of the digits of 10 = 1+0 = 1 which is the remainder when 10 is divided by 9.

Sum of the digits of 11 = 1+1 = 2 which is the remainder when 11 is divided by 9.

Sum of the digits of 12 = 1+2 = 3 which is the remainder when 12 is divided by 9.

Since the remainder is less than 9 when added for the first time, then the quotient will be the first digit of 11 which is 1 and the recurring part of the decimal number will be the remainder.

For example,. The sum of the digits of 16 is 7. Since 7 is less than 9, then the quotient is 1 because it is the first digit of 16 and the recurring part of the decimal is 7. Hence  = 1.7777777……………….

If we look at (vi) – (vii), the sum of the digits of the numerator is more than 9.

Sum of the digits of 47 = 4+7 = 11. In this case the digits of 11 must be added to get a single digit. So the sum of the digits of 11 = 1+1 = 2 which is the remainder when 47 is divided by 9 and the quotient will be the sum of first digit of 47 and 1 because the first remainder 11 is between 9 and 18 which will be 4+1 = 5.

Hence = 5.2222222222222222……………………………

Sum of the digits of 78 = 7+8 = 15. In this case the digits of 15 should be added to get a single digit number. So the sum of 15 = 1+5 = 6 which is the remainder when 78 is divided by 9 and the quotient will be the sum of the first digit of 78 and 1because the first remainder 15 is between 9 and 18, which will be 7+1 = 8.

Hence  = 8.666666666666666………………………………..

Division of a three digit number and 9.

= 13.6666666666666666666……………………..

= 46.777777777777777777……………………...

= 25.888888888888888888………………………

To find the value of (i) above, find the sum of the digits of 123 to get the remainder. So the sum of the digits of 123 = 1+2+3 = 6 and the quotient will be the sum of the first two digits of 123 which are 12 and the first digit of 123 which is1. Quotient = 12 + 1 = 13.

Hence  = 13.66666…………………………………………………….

To find , find the sum of the digits of 421 to get the remainder. So the remainder = 4+2+1 = 7. Quotient is equal to the sum of the first two digits of 421 which are 42 and the first digit of 421 which is 4. So quotient = 42+4 = 46.

Hence  = 46.77777777…………………………………………………...

(c )  To find , find the sum of the digits of 233 to get the remainder. So the remainder =

2+3+3 = 8. Quotient is equal to the sum of the first two digits of 233 which are 23 and

the first digit of 233 which is 2. So quotient = 23+2 = 25.

Hence  = 25.8888888888………………..

Division when the remainder is more than 9

= 40.444444444………………………..

= 58.5555555………………………

= 31.44444444………………………

= 84.33333333……………………

= 94.11111111……………………

To express (i) as a decimal number, first find the sum of the digits of 364 to get the remainder. The remainder = 3 + 6 + 4 = 13. Again add the digits of 13 to get a single digit remainder which is 1+3 = 4. To find the quotient since the remainder at first was more than 9, add 1 because 13 is between 9 and 18 to the first two digits of 364 which are 36 to get 36 + 1 = 37 and then add 37 to the first digit of 364 which is 3 to get the quotient. So the quotient = 37 + 3 = 40.

Hence  = 40.444444444…………………

To express (ii) as a decimal number, first find the remainder by adding the digits of 527. The remainder = 5 + 2 + 7 = 14. Again add the digits of 14 to get a single digit remainder. Sum of the digits of 14 = 1 + 4 = 5. To find the quotient since the remainder at first was more than 9, add 1 because 14 is between 9 and 18 to the first two digits of 527 which are 52 to get 52 + 1 = 53 and then add 53 to the first digit of 527 which is 5 to get the quotient. So the quotient = 53 + 5 = 58.

Hence  = 58.555555555……………..

To express (v) as a decimal number, first find the remainder by adding the digits of 847.

The remainder = 8 + 4 + 7 = 19. Again add the digits of 19 to get a single remainder. Sum of the digits of 19 = 1 + 9 = 10, again add the digits of 10. Sum of the digits of 10 = 1+0 = 1 which is the remainder. To find the quotient since the remainder at first was between 18 and 27, add 2 to the first digits of 847 which are 84 to get 84 + 2 = 86 and then add 86 to the first digits of 847 which is 8 to get the quotient. So the quotient = 86 + 8 = 94.

Hence  = 94.1111111111……………………

Application of 9 in the expression of recurring decimals as fractions

Division of a number which is a multiple of 9 divisible by 10.

= 0.07777777…………………

= 0.055555555………………..

= 0.02222222………………….

The three examples show that if the numerator is less than 10 and the denominator is 90, then the recurring figure will start after the zero on the right hand side of the point. This means that there will be a 0 in between the point and the recurring figure.

Division by 900

= 0.007777777………..

= 0.00555555………..

If the numerator is less than 10 and the denominator is 900, then the recurring figure will start after two zeros on the right hand side of the point.

The above patterns of the results of dividing a number by the multiple of 9 divisible by 10, 100, 1000………………,    can help to express recurring fractions into the form of   .

Express 0.04444444…………………,    as a fraction.

Since there is only one zero between the point and the recurring figure, and also one digit which is 4 recurring, then 0.044444….. =

Express 0.00343434………..as a fraction.

Since there are two zeros between the point and the recurring figures, and also two digits 34 recurring, then 0.00343434…. =

Express 0.00012121212……………,             as a fraction.

Since there are three zeros between the point and the recurring figures, and also two digits 12 recurring, then 0.000121212………. =

Express 3.1444444……………..as a fraction.

Round off 3.14444…… correct to 2 decimal places because there is only one figure between the point and 4 which is recurring. 3.144….. = 3.14, then split it into two parts. 3.14 = 3.1 + 0.04.

Express 3.1 and 0.04 as fraction bearing in mind that 4 is recurring.

3.1 =  and 0.04 =   and find the sum of the two fractions.

+     =

=  .

Hence 3.144444…….. =

Express 23.5678787878…………………,    as a fraction.

Round off 23.56787878………..correct to 4 decimal places because the two recurring figures 78 are after two decimal places. So 23.5678787878……… = 23.5678. Split 23.5678 into two parts. 23.5678 = 23.56 + 0.0078. Express 23.56 and 0.0078 as fractions and find the sum of the two fractions bearing in mind that 78 is recurring.

23.56 =  and 0.0078 =

+     =

=

Hence 23.5678787878…… =

The number 9 and multiplication.

Ordinary method                                                      Kwenge’s method

4 x 9 = 36                                                                 4 x 9 = 40 – 4

= 36

7 x 9 = 63                                                                  7 x 9 = 70 – 7

= 63

56 x 9 = 504                                                             56 x 9 = 560 – 56

= 504

478 x 9 = 4302                                                          478 x 9 = 4780 – 478

= 4302                                                                        = 4302

Multiplication of the number and the multiple of 9 with all the digits being 9s

(i)                12 x 9 = (12 x 10) – 12

= 120 – 12

= 108

(ii)              14 x 99 = (14 x 100) – 14

= 1 400 – 14

= 1386

(iii)            135 x 999 = (135 x 1000) – 135

= 135 000 – 135

= 134 865

If a and b are whole numbers and all the digits of b are 9s then a x b = (a x 10k) – a where k is number of 9s for b.

Multiplication of a number and the multiple of 9 with all digits being 9s

(i) 2 x 9 = 29 – (9+2)          (ii) 17 x 9 = 179 – (17 + 9)      (iii) 23 x 99 = 2399 – (23 + 99)

= 29 – 11                                = 179 – 26                                  = 2399 – 122

= 18                                         = 153                                        = 2277

(iv)     312 x 9999 = 3 129 999 – (312 + 9 999)

= 3 129 999 – 10 311

= 3 119 688

Formula; a x b = ab – (a+b) where ab is a and b brought together in the order they are written and b is the multiple of 9 and all the digits of b being 9s

Application of Kwenge’s combination theory of numbers

(i)     3 + 4 = 34 – (3x10 – 3 )                    (ii)    45 + 8 = 458 – (45x10 – 45 )

= 34 – 27                                                     = 458 – 405

= 7                                                               = 53

32 + 41 = 3241 – (32x100 – 32)           (iv) 27 + 123 = 27 123 – (27x1 000 – 27)

= 3241 – 3168                                                = 27 123 – 26973

= 73                                                                = 150

General formula: If a and b are whole numbers then a + b = ab – (a x 10k – a) where k is the number of digits of b.

The number 9 is unique because of the impact it has on other numbers in terms of addition, subtraction, multiplication and division. It is 9 and the numbers that have relationship with 9 that make division to produce recurring fractions or repeated decimals.

Generally, any number divided by the multiple of 9 with digits all being 9s or a multiple of 3 with digits all being 3s or a multiple of 6 with digits all being 6s will give the answer which is a repeating decimal. The results also show that application of the number 9 in solving mathematical problems can make learning and teaching of mathematics be interesting to the learners as well as to the teachers and its uniqueness can be applied in solving different problems such as expression of recurring decimals in form of fractions. The knowledge about the characteristics of 9 can help the teachers to be able to engage learners into the process of mathematical thinking (Kwenge, 2014). The number 9 is unique and has some powers and characteristics which other numbers do not have. To understand Kwenge’s method, it needs reasoning and thinking that can help the learners develop analytical and critical thinking and appreciate the role that different numbers are playing in problem solving. With the help of 3, 6 and 9 learners, teachers and mathematicians can understand why division of numbers not involving 3, 6 and 9, do not have same pattern. Understanding the pattern of repeated fractions or decimals can help man understand the reoccurrence of some natural phenomena that occur on our planet and in the universe as a whole. The special relationship that 9 has with other numbers in terms of addition, subtraction, multiplication and division is what is called Kwenge’s combination theory of numbers. The purpose of teaching mathematics is to enable learners to develop and use investigating patterns which will allow them to experience the excitement and satisfaction of mathematical discovery (Kwenge, Mwewa & Mulenga, 2015). Learning about the impact 9 has on other number in terms of addition, subtraction, division and multiplication can make learners develop the mathematical skills that can make them be risk-takers, inquirers and critical thinkers (Kwenge, 2014). The use of mathematical investigation skills gives learners the opportunity to apply mathematical knowledge and problem-solving techniques to investigate a problem, generate ideas or rules, analyse information, find relationships and patterns, describe these mathematically as general rules, and justify or prove them. Mathematics should be visualised as the vehicle to train learners to think, reason, analyse and articulate issues or problems logically. Apart from being a specific subject, mathematics should be treated as a concomitant to any subject involving analysis and reasoning (Government of India–Ministry of Human Resource Development, 1998). Using number patterns can help learners and teachers acquire skills in thinking, reasoning, analysing and articulating issues logically. The higher aim of teaching mathematics is to help the learner to develop the ability to think and reason mathematically, to pursue assumptions to their logical conclusion and to handle abstraction. It includes a way of doing things, and the ability and the attitude to formulate and solve problems (National Council of Education Research and Training, 2005). The results show that doing activities involving combination theory of the numbers can help the learners and the teachers of mathematics develop the ability to think creatively and mathematically (Brian, 1986). Creativity is about generating new and useful ideas and rules. Engagement in creativity would make learners be involved in producing something new and useful with respect to the previous knowledge. Creative work requires applying and balancing three abilities that can all be developed. The three abilities are; synthetic, the analytic, and the practical abilities: Synthetic ability is the ability to generate novel and interesting ideas (Sara & Moshe, 2015).

For learners have to learn best, they need to be exposed to the use of a wide range of teaching methods (Portman J and Richardson, 1997). They should be encouraged to explore the numbers so that they can discover the hidden powers in them and use them in solving the problems they are encountering in their everyday life and be able to understand the universe we live in. Teachers of mathematics should try by all means to use models and other teaching aids when teaching mathematics and engage learners in understanding the patterns which numbers form when carrying out the 4 basic operations of mathematics. Numbers being defined as ideas have unique characteristics which teachers of mathematics need to explore because whatever exists has elements of numbers. There is nothing that exists without numbers. Studying the number 9 is exciting to learners and can make learners develop interest in doing various mathematical activities.

In inductive reasoning conclusion is made from specific observations to broader generalisations and theories. Informally, it is sometimes called a "bottom up" approach. In inductive reasoning, the process of establishing the relationship begin with specific observations and measures, to detect patterns and regularities, formulate some tentative hypotheses that can be explored, and ending up by developing some general conclusions or theories. For Singh (2011), inductive reasoning is a process of reasoning which is used for developing more general rules from specific observations. The study is also guided by the theory of learning based on vision which according to Miller (1983) possess two stages of learning, the first stage being differentiation in which categorization and classification of information take place and the second stage being interpretation which involves synthesising of knowledge for making conclusions and judgments about the newly integrated information. For Lawler (1981), acquisition of mathematical knowledge requires deep understanding of procedural and conceptual knowledge and relationship of variables involved. Relationships can be established on two levels; understanding which originates from the ideas presented within the context and the relationships that are understood in an environment where appropriate abstractions have been made.