NUMBERS

A number is an arithmetic value used for representing quantity and used in making calculations.

We use the digits from 0 to 9 to form all other numbers.

0

1

2

3

4

5

6

7

8

9

With the help of these digits, we can create infinite numbers e.g. 10,11,50,19,10001,etc.

TYPE OF NUMBERS

Natural/Counting Numbers

These are numbers known as counting numbers that contain the positive integers from 1 to infinity.

The set of natural numbers denoted as "N" and it includes, N = {1,2,3,4,5,6,7,8,...}

Whole Numbers

These are known as non-negative integers and it does not include any fraction or decimal part.

It is denoted as "W" and the set of whole numbers includes, W = {0,1,2,3,4,5,6,7,8,9,10,...}

NOTE: Whole numbers include the integer, 0 but natural numbers do not include 0 since it starts from 1.

We can therefore say that set of natural numbers is a subset of set of whole numbers (can be found in the set of whole numbers).

Integers

These are the set of all whole numbers but it includes a negative set of natural numbers also. Hence whole numbers and natural numbers are subsets of set of integers.

"Z" represents integers and the set of integers include, Z = {...,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,...}

Real numbers

All the positive and negative integers, fractional and decimal numbers without imaginary numbers. Hence the set of natural numbers, whole numbers, integers, fractional and decimal numbers are all subsets of the set of real numbers.

Rational Numbers

Any number that can be written as a ratio of one number over another is known as rational numbers. This means that any number that can be written in the form a/b, where b is a non-zero number is a rational number.

Any fraction with non-zero denominators is a rational number. E.g. 1/4,2/3,6/9 and so on.

The number "0" is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3,0/4,0/5, etc.

The symbol "Q" represents the rational number.

Irrational Numbers

These are numbers that cannot be expressed as the ratio of one over another, thus no ratio can be written for that number.

In other words, we can say that irrational numbers cannot be represented as the ratio of two integers.

Examples include, pi(π=3.14159265…), √2, √3, √5, etc.

It is represented by the symbol "P".

Complex Numbers

These are numbers that can be written in the form of a+bi where "a" and "b" are the real number and the "i" is an imaginary number. E.g. 1+2i, where 1 is the real number and 2i the imaginary number

It is represented by the symbol "C".

Imaginary Numbers

These are complex numbers that can be written in the form of the product of a real number and the imaginary unit "i". Eg. 2i,3i,4i, etc.

Even Numbers

These are numbers exactly divisible by 2. These can be positive or negative integers.

Examples include 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100,...

All the multiples of 2 (positive and negative) are even numbers.

Odd Numbers

These are numbers which are not exactly divisible by 2. These can be both positive and negative integers.

All integers which are not in the set of multiples of 2 are odd numbers.

Examples include 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99,...

Prime Numbers

These are numbers that have only two factors,thus 1 and the number itself. These numbers can only be exactly divisible by 1 and the number itself.

Examples include 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,...

Composite Numbers

These are numbers that have more than two factors.

Examples include 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100,...

NOTE: 1 is neither a prime nor composite number

APPLICATION OF KNOWLEDGE

Question 1

Which of the following is the largest set?

A. {Whole numbers} B. {Composite} C. {Natural numbers} D. {Integers}

[B.E.C.E 2022 Section A Question 21]

Solution

Question 2

If P = {x:x is an even number greater than two and less or equal to twelve}, list the members of P

A. {2, 4, 2, 8, 10, 12} B. {3,4,6,8,10,12}

C. {2, 4, 6, 8, 10} D. {4, 6, 8, 10, 12}

[B.E.C.E 2018 Section A Question 2]

Solution

Question 3

E and F are subsets of the universal set U such that

U = {natural numbers less than 15}
E = {even numbers between 1 and 15}
F = {multiples of 4 between 9 and 15}

List the elements of U,E and F

[B.E.C.E 2008 Section B Question 1]

Solution

Question 4

If the set P = {1,2,3,4,5} which of the following statements best describes P?

A. Set of whole numbers up to 6

B. Set of counting numbers less than 6

C. Set of counting numbers greater than 6

D. Set of integers less than 6

[B.E.C.E 2005 Section A Question 5]

Solution

Question 5

U = {1,2,3,4,...,18};
A = {Prime numbers} and
B = {Odd numbers greater than 3}.

If A and B are subsets of the universal set, U, list the members of A and B

[B.E.C.E 2003 Section B Question 1]

NOTE: The list of the subsets should be elements in the universal set, U. Hence can't be more than 18

Solution

ADDITION OF NUMBERS

Addition in math is a process of combining two or more numbers. Addends are the numbers being added, and the result or the final answer we get after the process is called the sum.

The addition symbol used to indicate addition of numbers is "+", called the plus symbol

We can the numbers vertically. Here, we arrange the numbers vertically using their respective place values, like ones, tens, hundreds, thousands, etc. We start adding from the right side (or digit at ones or unit place).

To add numbers, you can follow the steps below:

Step 1

Write the numbers one below the other as per the places of the digits. It is better to arrange by the number of digits. The number with the highest digits comes first followed by the next and so on.

Step 2

Start adding from the digits in ones place. Write the sum under the ones digit.

If the resut is 10 or more (two digits), write the first digit (the ones) and carry-over the second digit (the tens)

Step 3

Add the tens digits, the hundreds, the thousands and so on.

If the resut is 10 or more (two digits), write the first digit (the ones) and carry-over the second digit (the tens)

Place value chart

ILLUSTRATIONS

Example 1

Add 34 and 52

Step 1

Write the numbers one below the other as per the places of the digits

Step 2

Start adding from the digits in ones place. Write the sum under the ones digit.

Step 3

Add the tens digits, the hundreds, the thousands and so on.

34 + 52 = 86

Example 2

Add 29 and 46

Step 1

Write the numbers one below the other as per the places of the digits

Step 2

Start adding from the digits in ones place. Write the sum under the ones digit.

If the resut is 10 or more (two digits), write the first digit (the ones) and carry-over the second digit (the tens)

9 + 6 = 15, the ones/first digit (5) is written and the tens/second digit is carried-over

Step 3

Add the tens digits, the hundreds, the thousands and so on and their corresponding carry-overs.

29 + 46 = 75

Example 3

Add 3475 and 2865

Explanation

Step 1

5 + 5 = 10

The ones (0) part is written and the tens digit, carry-over (1) is taken to the next digit

Step 2

1 + 7 + 6 = 14

The ones (4) part is written and the tens digit, carry-over (1) is taken to the next digit

Step 3

1 + 4 + 8 = 13

The ones (3) part is written and the tens digit, carry-over (1) is taken to the next digit

Step 4

1 + 3 + 2 = 6

3475 + 2865 = 6340

SUBTRACTION OF NUMBERS

The operation or process of finding the difference between two numbers or quantities is known as subtraction. To subtract a number from another number is also referred to as 'taking away one number from another'. The symbol for subtraction is "-".

Terms in subtraction

Minuend: A minuend is the number from which the other number is subtracted.

Subtrahend: A subtrahend is the number which is to be subtracted from the minuend.

Difference: A difference is the final result after subtracting the subtrahend from the minuend.

To subtract one number from the other, follow the steps below:

Step 1

Write the numbers one below the other as per the places of the digits.

Step 2

Start subtracting from the digits in ones place. Write the difference under the ones digit.

If the subtrahend is larger than the minuend, borrow 1 from the next digit which becomes 10 when adding to the preceeding digit and so on.

Step 3

Subtract the tens digits, the hundreds, the thousands and so on.

If the subtrahend is larger than the minuend, borrow 1 from the next digit which becomes 10 when adding to the preceeding digit and so on.

ILLUSTRATIONS

Example 1

Subtract 3248 from 6589

Example 2

Subtract 19 from 31

Step 1

Write the numbers one below the other as per the places of the digits.

Step 2

Start subtracting from the digits in ones place. Write the difference under the ones digit.

If the subtrahend is larger than the minuend, borrow 1 from the next digit which becomes 10 when adding to the preceeding digit and so on.

9 is larger than 1 hence 1 is borrowed from 3 which becomes 10 and its added to the 1 at the ones to give 11.

The 3 at the tens becomes 2 since 1 has been borrowed.

9 is then subtracted from the 11 giving 2

1 is then subtracted from the 2 to give 1

31 - 19 = 12

Example 3

Subtract 25632 from 48756

Example 4

Subtract 3678 from 8162

Explanation

Step 1

8 is larger than 2, hence 1 is borrowed from 6 and its added to 2 which becomes (10+2 = 12) and 8 can now be subtracted from the 12 which gives 4

The 6 that the one is borrowed from now becomes 5

Step 2

7 is larger than 5, hence 1 is to be borrowed from the next digit (1) which becomes (10+5 = 15) and 7 can now be subtracted from the 15 which gives 8

Step 3

When 1 is borrowed from the 1, it becomes 0. 6 is greater than 0, hence 1 is borrowed from 8 which becomes (10+0 = 10) and 6 can now be subtracted from the 10 which gives 4

Step 4

The 8 becomes 7 when 1 is borrowed. The 3 is now subtracted from the 7 which gives 4

8162 - 3678 = 4484

MULTIPLICATION OF NUMBERS

Multiply means the repeated addition of groups of equal sizes. The symbol for multiplication is "x"

When we multiply two numbers, the answer is called product. The number of objects in each group is called multiplicand, and the number of such equal groups is called the multiplier

To list the multiplications of a number, you have to repeatedly add the number to get each terms

EXAMPLES

Multiples of 2

2	x	1	=	0+2	=	2
2	x	2	=	2+2	=	4
2	x	3	=	4+2	=	6
2	x	4	=	6+2	=	8
2	x	5	=	8+2	=	10
2	x	6	=	10+2	=	12
2	x	7	=	12+2	=	14
2	x	8	=	14+2	=	16
2	x	9	=	16+2	=	18
2	x	10	=	18+2	=	20
2	x	11	=	20+2	=	22
2	x	12	=	22+2	=	24
2	x	13	=	24+2	=	26
2	x	14	=	26+2	=	28
2	x	15	=	28+2	=	30

Thus the multiples of 2 are 2,2+2 = 4,4+2 = 6,6+2 = 10, 10+2 = 12, 12+2 = 14, 14+2 = 16, 16+2 = 18, 18+2 = 20, 20+2 = 22, 22+2 = 24, 24+2 = 26, 26+2 = 28, 28+2 = 30,...

Multiples of 3

3, 3+3 = 6, 6+3 = 9, 9+3 = 12, 12+3 = 15, 15+3 = 18, 18+3 = 21, 21+3 = 24, 24+3 = 27, 27+3 = 30, 30+3 = 33, 33+3 = 36,...

You can use this step of obtaining the multiples of a number to write down the multiplication tables that you have difficulty in memorizing on the question paper before solving the examination questions so that you can quickly make reference to them in case you are not allowed to carry calculator to the examination hall.

While solving multiplication problems, one-digit numbers can be multiplied in a simple way by using the multiplication tables, but for larger numbers, we split the numbers into columns using their respective place values, like ones, tens, hundreds, thousands, and so on.

Type of multiplication problems

1. Multiplication without regrouping
2. Multiplication with regrouping

Multiplication without regrouping

Multiplication of two numbers without regrouping involves smaller numbers where there is no need to take a carry-over to the next higher place value. It is the basic level that could help a learner understand the basics of multiplication before moving on to the higher level of problems including regrouping.

Steps

Multiply each of the digits starting from the ones by the multiplier

NOTE: When the multiplier has more than one digits (ones, tens, hundreds, ...), after the multiplication of each digit, you start with zeros at the ones, tens,hundreds, etc before multiplying the subsequent digits.

Once you are done with the multiplication of each of the digits of the multiplier, you can sum their products to obtain the final answer.

Example 1

Multiply 42312 by 2

TTh	Th	H	T	O
4	2	3	1	2
			x	2
8	4	6	2	4

Explanation

Step 1

Multiply the ones part(2) by 2 = 2 x 2 = 4 and write it below

Step 2

Multiply the tens part(1) by 2 = 1 x 2 = 2 and write it below

Step 3

Multiply the hundreds part (3) by 2 = 3 x 2 = 6 and write it below

Step 4

Multiply the thousands part(2) by 2 = 2 x 2 = 4 and write it below

Step 5

Multiply the ten thousands part(4) by 2 = 4 x 2 = 8 and write it below

Multiplication with regrouping

In this type of multiplication, we need to take a carry-over to the next higher place value. Thus the product gives more than one digit, hence the ones part (first digit) is written and the tens part (second digit) is carried over.

Example 1

Find the product of 2468 and 8

Th		H	T	O
	3	5	6
	2	4	6	8
			x	8
1	9	7	4	4

Explanation

Step 1

8 x 8 = 64, we write down the ones part (4) and carry-over the tens part(6)

Step 2

6 x 8 = 48. There is already carry-over of 6, hence we add the 6 to the 48, 48+6 = 54. we write down the ones part (4) and carry-over the tens part(5)

Step 3

4 x 8 = 32. There is already carry-over of 5, hence we add the 5 to the 32, 32+5 = 37. we write down the ones part (7) and carry-over the tens part(3)

Step 4

2 x 8 = 16. There is already carry-over of 3, hence we add the 3 to the 16, 16+3 = 19. Since that's the last part we write down the result (19).

Example 2

Find the product of 1021 and 32

	Th		H	T	O
		1	0	2	1
			x	3	2
		2	0	4	2
+	3	0	6	3	0
	3	2	6	7	2

Explanation

Multiplication by the ones part (2) of the 32 multiplier

Step 1

1 x 2 = 2

Step 2

2 x 2 = 4

Step 3

0 x 2 = 0

Step 4

1 x 2 = 2

Multiplication by the tens part (3) of the 32 multiplier

Step 5

Since that's the second digit, start with zero (0)

Step 6

1 x 3 = 3

Step 7

2 x 3 = 6

Step 8

0 x 3 = 0

Step 9

1 x 3 = 3

Step 10

Now that we have multiplied all the digits of the multiplier with the multiplicand, we will add the digits obtained in a vertical manner

2042 + 30630 = 32672

Example 3

Find the product of 1025 and 34

Factors

A factor is a number that divides another number, leaving no remainder. In other words, if multiplying two whole numbers gives us a product, then the numbers we are multiplying are factors of the product because they are divisible by the product.

For example 2 x 3 = 6, hence 2 and 3 are factors of 6

Thus factors of a number are the list of numbers which can exactly divide that number without a remainder. Every number has a factor of 1 and itself since one divides all numbers without a remainder and the number can divide itself as well without a remainder.

Factors of 2

{1,2}

Factors of 3

{1,3}

Factors of 4

{1,2,4}

Factors of 5

{1,5}

Factors of 6

{1,2,3,6}

Factors of 7

{1,7}

Factors of 8

{1,2,4,8}

Factors of 9

{1,3,9}

Factors of 10

{1,2,5,10}

Verifying if all factors has been listed

1. List the factors in ascending order (from the lowest to the highest)

2. Multiply the first by the last to see if the result gives you the number whose factor you are listing

3. Multiply the second by the last but two to see if the result gives you the number whose factor you are listing

4. You continue in that order until all the numbers paired with their product giving the number whose factor you are listing

NOTE: If the square of a factor gives the number whose factor you are listing, the number is paired by itself

Illustration

List of factors of 30 are {1,2,3,5,6,10,15,30}

You can verify if you are not missing a factor as follows

1 x 30 = 30

2 x 15 = 30

3 x 10 = 30

5 x 6 = 30

If there is an element left out, then it's pair is missing. Unless the square of that number gives the number we are looking for its factor

List of factors of 36 are {1,2,3,4,6,9,12,18,36}

You can verify if you are not missing a factor as follows

1 x 36 = 36

2 x 18 = 36

3 x 12 = 36

4 x 9 = 36

6 x 6 = 36

As you can see, during the pairing, 6 will be left alone but it's square gives the number whose factor is to be listed, hence the list is complete

If one of the numbers, doesn't have a pair and it's square doesn't give the number whose factor you are looking for, then you are missing a factor which you have to figure out.

DIVISION

Division is a mathematical operation which involves the sharing of an amount into equal-sized groups. For example, "8 divided by 2" means "8 shared into 2 equal groups", which would be 4.

In mathematics, this is written with the division symbol (÷), called an obelus.

Division can also be presented as a fraction which is represented as a forwards slash (/)

Thus 8 divided by 2 can be represented as 8÷2 or 8/2

The first number (the amount being divided) is called the dividend and the second number (whatever the dividend is being divided by), is called the divisor and the answer is called the quotient.

Thus in 8/2, the 8 is the dividend and the 2 is the divisor and the answer which is 4 is the quotient

Remainders

Sometimes a number cannot be shared into equal groups with an integer (whole number) as an answer, so we are left with a remainder.

For instance when dividing 11 by 2, there will be a remainder of 1 since we can split it into 5 equal groups and there will be 1 left over

Thus 11/2 = 5 remainder 1

Long Division

Long division is the mathematical method for dividing large numbers into smaller groups or parts. It helps to break down a problem into simple and easy steps.

The long divisions have dividends, divisors, quotients, and remainders.

Steps for dividing numbers using long division

There are five steps to solve every long division problem with ease.

1. Divide
2. Multiply
3. Subtract
4. Bring down
5. Repeat or Remainder

Explanation to the 75/4

1. Divide

4 can go into 7, 1 time, hence the 1 is written at the quotient side.

2. Multiply

The 1, the quotient is then multiplied by the 4, the divisor and the result (4) is written down.

3. Subtract

The 4 is subtracted from 7 and the result (3) is written down together with the other remaining dividend (5) to become 35.

The steps are repeated for the new dividend (35)

Repeating step for dividend 35

1. Divide

4 can go into 35, 8 times (4 x 8 = 32), hence the 8 is written at the quotient side.

2. Multiply

The 8, the quotient is then multiplied by the 4, the divisor and the result (32) is written down.

3. Subtract

The 32 is subtracted from the 35 and the result is written down

There will be a remainder of 3

Hence 75/4 = 18 remainder 8

Simplifying the remainder into decimals

If the dividend is lesser than the divisor, you can add a decimal point to the quotient and add 0 to the dividend to continue with the simplification.

If the dividend is greater than the divisor, after adding the first decimal point, another other 0 you add doesn't add any point to the quotient.

If the dividend is lesser than the divisor, after the decimal point (0.), any time you add a zero, you also add zero after the decimal until the dividend is divisible by the divisor

Explanation from the remainder 3

Dicimal point added to quotient

A decimal point is added to the quotient and 0 is added to the remainder 3 which becomes 30

Divide 30 by 4

4 goes into 30, 7 times (4 x 7 = 28), the 7 is written at the quotient after the decimal point

Multiply 7 by 4

The 7 at the quotient is multiplied by the divisor 4 and the result (28) is written down.

Subtract

The 28 is then subtracted from the 30 and the result (2) is written down

Add 0 to the 2

Since 4 can't go into 2, 0 is added to 2 to become 20

Divide 20 by 4

20 is divided by 4 and the result (5) is written at the quotient

Multiply 5 by 4

The 5 at the quotient is multiplied by the 4 and the result (20) is written down

Subtract 20 from 20

20 is subtracted from 20 which gives 0 (no remainder), hence the long division is completed

Examples

Divide 3 by 20

Explanation

Write 0. and add 0 to 3

Since 20 can't divide 3, write 0. at the quotient and add 0 to the dividend (3) to become 30

Divide 30 by 20

20 goes into 30, 1 time, and its written at the quotient

Multiply 1 by 20

The 1 at the quotient multiplies the divisor (20) and the result, 20 is written down

Subtract the 20 from 30

The 20 is subtracted from 30 and the 10 obtained is written down

Add 0 to the 10

Since 20, can't divide 10, 0 is added to the 10 to become 100

NOTE: Since we've already obtained our first divisible dividend, we don't have to add 0 to the quotient any time we add 0 to the dividend

Divide 100 by 20

20 goes into 100, 5 times and it is written at the quotient

Multiply the 5 by 20

5 at the quotient multiplies the divisor (20) and the 100 obtained is written down.

Subtract 100 from 100

The subtraction of 100 from 100 gives 0, hence the long division is completed.

Divide 2 by 500

Explanation

Write 0. and add 0 to the 2

0. is written at the quotient and 0 is added to the 2 to become 20

500 cannot divide 20 yet so 0 is added after the decimal point and another 0 is added to the dividend (20) to become 200

500 cannot divide 200 yet so another 0 is added to the quotient and another 0 is added to the dividend (200) to become 2000

Divide 2000 by 500

500 divides 2000, 4 times, hence 4 is written at the quotient

Multiply 4 by the 500

The 4 is multiplied by the 500 to obtain 2000 and its written below

Subtract 2000 from 2000

The subtraction of 2000 from 2000 gives 0, hence the long division is completed.

APPLICATION OF KNOWLEDGE

Question 6

Find the missing addend:

A. 8393 B. 1967 C. 2512 D. 467

[B.E.C.E 1990 Section A Question 18]

Solution

Question 7

Find the missing addend:

A. 1300 B. 1720 C. 2765 D. 4065

[B.E.C.E 1991 Section A Question 3]

Solution

Question 8

If X = {prime numbers less than 13} and
Y = {odd numbers less than 13}

(i) List the members of X and Y

[B.E.C.E 1991 Section B Question 1]

Solution

Question 9

The following addition is done in base ten. What number represents a b c?

A. 324 B. 342 C. 423 D. 435

[B.E.C.E 1996 Section A Question 2]

Solution

Question 10

Which of the following is the set of factors of 12?

A. {12,6,4,3,2,1} B. {12,6,4,3,2}

C. {12,6,4,2} D. {6,4,2,1}

[B.E.C.E 1994 Section A Question 1]

Solution

Question 11

Find the set of prime factors of 12.

A. {3} B. {2,3}

C. {3,4} D. {2,6}

[B.E.C.E 1992 Section A Question 1]

Solution

Question 12

Find the missing numbers in the set

{5,10,15,...,25,...,...,40}.

A. 20 and 30. B. 30 and 35.

C. 20, 35 and 40. D. 20,30 and 35.

[B.E.C.E 2001 Section A Question 1]

Solution

Question 13

Find the sum of all the even numbers between 70 and 80.

A. 200 B. 223 C. 280 D. 300

[B.E.C.E 2001 Section A Question 7]

Solution

Question 14

M is a set consisting of all positive integers between 1 and 10. P and Q are subsets of M such that P = {factors of 6} and Q = {multiples of 2}

(i) List the elements of M, P and Q.

[B.E.C.E 2001 Section B Question 1 a]

Solution

Question 15

Find the product of 17 and 121.

A. 968 B. 1,751 C. 2057 D. 8,591

[B.E.C.E 2003 Section A Question 9]

Solution

Question 16

A and B are subsets of a universal set U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}

Such that A = {even numbers}
B = {multiples of 3}.

(i) List the elements of A,B

[B.E.C.E 2010 Section B Question 1 b]

Solution