KNOWLEDGE OF NUMBERS

NUMBERS

Def. Number is the sign or word that represents the amount or quantity e.g. one, two, three etc. there are symbols that used to represent numbers, those symbols are called numerals e.g. 1, 2, 3 ….. In definition numerals are symbols used to represent numbers.

BASE TEN NUMERATION

The current forms of symbols which are now used to represent numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 they were brought and discovered by Hindus and are called Hindu – Arabic Numerals. They are only ten.

Symbols that appear in numbers are called digits, the word Digit derived from a Latin word “digitus” which means “finger”. Consider the number 6275. This number has four digits, which are 6, 2, 7 and 5

PLACE VALUE

As discussed earlier that each number has digits, here we discuss about place value what is it? The answer is:-

Each digit has its value according to its position in number. Consider the number 6275 starting discovering from right, the right digit has the lowest value but as you go to the left the digit value increase from 1 to 10 to 100 to 1000 to 10000 to 100000 etc. but we don’t pronounce as normal like ten, hundred but they are pronounced as follows:-

1                      -           Ones

10                    -           Tens

100                  -           Hundreds

1000                -           Thousands

10000              -           Ten Thousands

100000                        -           Hundred Thousands

1000000          -           Millions

The table below shows the number 6275893 and the place value of each digit.

6	2	7	5	8	9	3
Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Ones

From the above each digit has the place value for example number 5 occupies the thousands, so it can be written as 5 x 1000, 3 occupies ones so it can be written 3 x 1

Example:

1.  25478 = 2x 10000 + 5x1000 + 4x100 + 7x10 + 8x1

2.  195 = 1x100 + 9x10 + 5x1                                             

3.  53 = 5x10 + 3x1

This method is called expanded form of the numeral

The method of writing numbers in groups of ten is called base ten numeration or decimal system numeration

Questions:

1.         Write down the place value of number 3 in the following numbers

            a) 6532                        b) 13                c) 863754        d) 377              e) 67543

2.         Write the following numbers in expanded form

            a) 14552          b) 685              c) 776512        d) 9012                        e) 23

3.         Write each of the following as a single number

            a) 3 x 1000 + 6 x 100 + 7 x 10 + 1 x 1

            b) 8 x 10000 + 2 x 1000 + 7 x 100 + 0 x 10 + 4 x 1

            c) 6 x 100 + 1 x 10 + 5 x 1

            d) 3 x 1000000 + 0 x 100000 + 3 x 10000 + 5 x 1000 + 2 x 100 + 7 x 10 + 9 x 1

4.         Write down five numbers that can be obtained from the digit 3, 6, 1, 8, 7

5.         How many digits do the following numbers have?

            a) 7924729

            b) 839 

            c) 848237129

            d) 8883

            e) 9494

NUMBERS UP TO ONE BILLION

In writing or reading very large number, it is advisable to separate the number three digits from left in order to understand it easily for example the number 267327236 can be written as 267,327,236.

HOW TO READ THE NUMBERS OR WRITE IN WORDS

As we discussed how to separate three digits form left to right, each group of separated digits has its value that make us easy to read or write it words see the table  below that shows the groups of number 267327236.

267	327	236
millions	Thousands	Hundreds

Reading this number is easy; first each number in group should be read as it is, example the first group is 267 that should be read as two hundred and sixty seven then check the value of the group, 267 is in millions so after writing it as two hundred and sixty seven then add millions, this will become two hundred and sixty seven millions

Now, we are going to see number 327 which is the second group of our number, reading it is as that we read above in millions, we first read it as it is, that will be three hundred and twenty seven then the group value of this number is thousands so we add the word thousands after three hundred and twenty seven and becomes three hundred and twenty seven thousands.

Using the principles above the last group will be written as two hundred and thirty six hundred

Then in general the whole number is pronounced or written in word as two hundred and sixty seven millions three hundred and twenty seven thousands two hundred and thirty six hundred

Numerals	The same numerals in words
34	Thirty four
178	One hundred and seventy eight
7,743	Seven thousands seven hundred and forty three
10,341	Ten thousands three hundred and forty one
42,855,132	Forty two millions eighty hundred and fifty five one hundred and thirty two

Questions:

Write the following numerals in words

1)  1,455          2) 7,763,522      3) 7,568         4) 134        5) 761, 887, 600

Write the following numbers in numerals

1) Seven hundred and forty five million eight hundred and fifty seven thousand one hundred and thirty nine.

2) Four thousand seven hundred and thirty two

3) Five hundred and six

4) Two million and five hundred

5) Nine million seven hundred and forty eight hundred and six

NATURAL AND WHOLE NUMBERS

When we count numbers from one (1) to infinity i.e. 1, 2, 3, 4, 5…..  In mathematics when you put dots at the end it indicates that the numbers continue with no end. Ok such numbers are called natural numbers. In definition natural numbers are numbers counted from 1, e.g. 1, 2, 3, 4, 5, 6, 7, 8, 9 …… These numbers are sometimes known as counting numbers and they are represented by N.

(i)      Natural numbers on the number line   

   1       2          3          4          5

There is another group of numbers which start from 0, e.g. 0, 1, 2, 3, 4, 5, 6, 7, 8 ……. These numbers are called whole numbers.

Whole numbers are the numbers which start from 0, e.g. 0, 1, 2, 3, 4, 5, 6, 7, 8 …….  And they are represented by W.

(i)      Whole numbers on the number line   

   0       1          2          3          4

Note:

All natural numbers are whole numbers but not all whole numbers are natural numbers.

Arrows in both number lines indicate that numbers continue with no end.

These are not only group of numbers in mathematics, others will be discussed later.

Even, odd and Prime numbers

Even number:

Even number is any integer which is exactly divisible by 2, e.g. 2, 4, 6, 8, 10, 18, 100, 150 etc.

Odd number:

Odd number is any integer which is not exactly divisible by 2, e.g. 1, 3, 5, 7, 11. 153, 37 etc.

Odd number is an opposite of even number, so if any number isn’t divisible by 2 that number is exactly odd number.

Prime number:

Prime number is any number that is not divisible by 3 and 7 e.g. 11, 13, 17, 43, 113 etc. to find a prime number you should remove all even numbers and numbers containing 5 in the place value of ones such as 25, 115, 45, 305, 245 etc. then after removing those, you should test the remaining to make sure that they are not divisible by both 3 and 7. Note that if it is not divisible by 3 but divisible by 7 that number is not prime number because the principle here is that should not be divisible by both and not only one. Example number 57, this number is not divisible by 7 but divisible by 3 so this is not a prime number.

Note:

The first four numbers before ten are prime numbers even if they are divisible by 3 and 7 those numbers are 2, 3, 5 and 7 so in listing prime numbers from the beginning those numbers should be included.

Examples:

(i) List all even numbers between 10 and 20.

Solution:

Step one, list all numbers between 10 and 20 but exclude 10 and 20, this becomes

11, 12, 13, 14, 15, 16, 17, 18 and 19

Step two, to get even numbers you should write those exactly divisible by 2, these are:-

12, 14, 16 and 18.

Therefore the answer is 12, 14, 16 and 18.

(ii) List all odd numbers between 10 and 20.

Solution:

Step one, list all numbers between 10 and 20 but exclude 10 and 20, this becomes

11, 12, 13, 14, 15, 16, 17, 18 and 19

Step two, to get odd numbers you should write those exactly not divisible by 2, these are:-

11, 13, 15, 17 and 19.

Therefore the answer is 11, 13, 15, 17 and 19.

(iii) What are the prime numbers between 25 and 34?

Solution:

Step one, list all numbers between 25 and 34 but exclude 25 and 34, this becomes

26, 27, 28, 29, 31, 32, 33.

Step two, remove all even numbers and that with 5 in ones position, and this becomes:-

27, 29, 31, 33.

Step three, the remaining should be tested if each number is divisible by 3 or 7, and we need the number which is not divisible by any of 3 and 7. Now let’s check it out:-

27 is divisible by 3                                (this is not prime number)

29 is not divisible by any of 3 and 7     (this is prime number)

31 is not divisible by any of 3 and 7     (this is prime number)

33 is divisible by 3                               (this is not prime number)

Therefore we only got two numbers which are not divisible by both 3 and 7, those numbers are 29 and 31

Therefore the answer is 29 and 31.

(iv) Which number is even and prime?

Solution:

The number which is even and prime is 2

QUESTIONS

(1) What do you understand by:

             i.  Natural and whole numbers

ii.  Even numbers

iii. Odd numbers

iv. Prime numbers

(2) Write number having 5 in hundreds position, 4 in thousands position, 0 in ones position and 4 in tens position.

(3) List all odd numbers between 60 and 70

(4) What are prime numbers between 1 and 20?

(5) Write the sum of all even numbers between 100 and 110.

(6)  List all odd natural numbers up to 10 on the number line.

(7)  Show prime numbers between 30 and 40 on the number line.  

(8) State whether the sum of the following is an even or odd

       i. When adding any two odd numbers

      ii. When adding any two even numbers

     iii. When adding any odd number and any even number.

(9) Is the product of any two even numbers even or odd?

(10)  Write down the first six prime numbers.

Note on the operation of whole numbers:

·         When adding two numbers the result is called “sum or total” example the sum of 2 and 4 is 6

i. e. (2 + 4 = 6).

·         When subtract two numbers the answer is called “the difference”  example the difference of 10 and 6 is 4  i.e. (10 – 6 = 4)

·         When multiply two numbers the answer is called “ the product” example the product of 3 and 5 is 15

i.e. (3 x 5 = 15) here 3 is multiplier, 5 is multiplicant and 15 is the product.

·         When dividing two numbers the result is called “the quotient” example 45 ÷ 5 = 9, here 45 is dividend, 5 is divisor and 9 is the quotient.

PROPERTIES OF WHOLE NUMBERS

In adding or multiplying whole numbers the following properties can be observed, let a and b be any whole number:-

Commutative property, a + b = b + a or a x b = b x a

                                    Example: 4 + 2 = 2 + 4 = 6 or 5 x 3 = 3 x 5 = 15

Associative property, (a + b) + c = a + (b + c) or (a x b) x c = a x (b x c)

                                    Example: (2 + 3) + 6 = 2 + (3 + 6) = 11 or (2 x 3) x 6 = 2 x (3 x 6) = 36

Distributive property, a x (b + c) = (a x b) + (a x c)

                                    Example: 2 x (3 + 4) = (2 x 3) + (2 x 4) = 14.

Note:

You can write a x (b + c) in short as a(b + c) because a number outside the brackets indicates the multiplication process.

Closure property, this occurs when dealing with addition or multiplication of whole numbers, where the result is also whole numbers. It is called closure property.

Example: 3 + 4 = 7, 3 x 4 = 12

               Therefore a + b = whole number

                                  a x b = whole number

Identity property, when a number let’s say “a” is operated by a certain number and the result is the same number “a” then the property is identity and the identity for the operation is that operated number

Example: a + 0 = a here the identity for operation is 0

               a x 1 = 1 here the identity for operation is 1

Inverse property, if two numbers are operated and leaves the answer 0 then this property is called inverse because one number is operated by its additive inverse which is the negative of that number. E.g. the additive inverse of a is –a and when operated “a – a = 0”

QUESTIONS:

1.  Using properties of whole numbers, give reasons why the following pairs are considered to be equal.

      (a) 3 + (4 + 5) = (3 + 4) + 5

      (b) (4 x 5) x 7 = 4 x (5 x 7)

      (c) (5 + 1) x 6 = (5 x 6) + (6 x 1)

2.  Express the following in the form of a x (b + c)

     (a) 6 x 1 + 6 x 5

     (b) 5 x 9 + 3 x 9

     (c) 5 x 3 + 5 x 9

3. Express each of the following in the form of a x b + a x c

     (a) 2(3 + 4)

     (b) 3 x (7 +2)

     (c)  12(3 + 1)

4. State the property that justify each of the following properties

     (a) 5 + -5 = 0

     (b) 3 (4 + m) = (3 x m) + (3 x 4)

     (c) d x 5 = 5 x d

FACTORS OF NATURAL NUMBERS:

Consider the number 10, any number that can divide it exactly is called the factor of that number 10, for example number 2, this is the factor of that number 10 because it can divide it exactly and get 5 so 2 is the factor of number 10. Other factors of 10 are 1, 5 and the same 10.

In general, factors of 10 are 1, 2, 5 and 10.

Characteristics of factors of natural numbers:

1.  Number one (1) is the factor of every number, when listing factors, number one should start.

2.  The last factor of each number is the same given number example when listing factors of 16, you should begin with 1 then the last factor should be 16.

3.  factors of natural numbers are less than the given number, you can’t get the factor which is greater than given number for example if you are ordered to find the factors of 8 and in listing you write the number which is greater than 8 then the answer is totally wrong.

Examples:

1.  List all factors of 20.

Answer:

Factors of 20 are: 1, 2, 4, 5, 10 and 20.

2.  What are the factors of 16?

Answer:

Factors of 16 are: 1, 2, 4, 8 and 16.

Prime factors:

As we discussed earlier about prime numbers, we all now understand the first prime numbers such as 2, 3, 5, 7 and so on.

Numbers are obtained from the product of these numbers, for example number 10 is obtained by the product of prime numbers 2 and 5 which is 2 x 5 = 10. So the prime factors of 10 are 2 x 10.

How to get prime factors:

Example:  write 120 as the product of prime numbers.

Solution:

You should begin dividing it by first prime number if it exactly divides it then put it in your table but if it can’t jump to another prime number, check it out below.

2    120            Here we put number two because it exactly divide 120.

2      60            After dividing the answer is 60 then we wrote it down there and we continue dividing it by 2

2      30            After dividing the answer is 30 then we wrote it down there and we continue dividing it by 2

3      15            After dividing the answer is 15 then we wrote it down there but we jumped to 3 because 2 can’t divide 15.

5        5            After dividing the answer is 5 then we wrote it down there but we jumped to 5 because 3 can’t divide 5

            1            The last we obtained 1 and that is the end of our solution.

Then by taking the left side prime numbers we get 2 x 2 x 2 x 3 x 5 = 2³ x 3 x 5

Therefore 120 = 2³ x 3 x 5

Example 2: write 90 as the product of prime numbers.

Solution:

Do it as example 1 above:

2   90

3   45

3   15

5     5

       1

Then by taking the left side prime numbers we get 2 x 3 x 3 x 5 = 2 x 3² x 5

Therefore 90 = 2 x 3² x 5

QUESTIONS:

1.  List all factors of the following numbers

     (a) 34          (b) 28        (c) 56          (d) 120             (e) 66

2.  Write each of the following number as the product of prime factors

     (a) 100        (b) 34        (c) 76          (d) 4200           (e) 180

3. Which number is the factor of every number?

4. List factors of the numbers whose their product represented by prime numbers below:

    (a) 2 x 3 x 5 (b) 2 x 2 x 2 x 5           (c) 3 x 3 x 3 x 5           (d) 2 x 3 x 5 x 5           (e) 2 x 2 x 2 x 2 x 3

Highest Common Factor (HCF) / Greatest Common Factor (GCF)