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ARITHMETIC FUNDAMENTALS: CONCEPTS AND RULES
TOPICS TO COVER
- NUMBERS
- INTEGERS - OPERATIONS OF WHOLE NUMBER
- ADDITION - SUBTRACTION - MULTIPLICATION - DIVISION
- WHOLE NUMBER LONG DIVISION
- FRACTION
- TYPES OF FRACTIONS
- PROPER FRACTION - IMPROPER FRACTION -MIXED FRACTION
- OPERATION IN FRACTIONS (ADDITION AND SUBTRACTION, MULTIPLICATION, DIVISION).
- DECIMAL POINT
- PRIME NUMBER
- COMPOSITE NUMBER
- A FACTOR
- DIVISIBILITY TESTS TABLE
- LOWEST COMMON MULTIPLE AND HIGHEST COMMON FACTOR
- DECIMAL PLACE VALUE
- OPERATIONS ON DECIMAL
- BODMAS
- PERFECT SQUARE AND PERFECT SQUARE ROOT
- RULES OF ADDITION AND SUBTRACTION
- BASIC RULES OF MULTIPLICATION AND DIVISION OF MATHEMATICAL SIGN/OPERATION
- LOWEST COMMON MULTIPLE
- HIGHEST COMMON FACTOR
- STANDARD FORM
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NUMBERS
THE INTEGERS
The most common numbers are those used for counting, namely the numbers
1, 2, 3, 4, …,
which are called the positive integers. Even for counting, we need at least one other number, namely, 0 (zero).
The positive integers and zero can be represented geometrically on a line, in a manner similar to a ruler or a measuring stick
For convenience, it is useful to have a name for the positive integers together with zero, and we shall call these the natural numbers. Thus 0 is a natural number, so is 2, and so is 124,521.
The natural numbers can be used to measure distances, as with the ruler. By definition, the point represented by 0 is called the origin. The natural numbers can also be used to measure other things. For example, a thermometer is like a ruler which measures temperature. However, the thermometer shows us that we encounter other types of numbers besides the natural numbers, because there may be temperatures which may go below 0. Thus we encounter naturally what we shall call negative integers which we call minus 1, minus 2, minus 3, . . . , and which we write as -1 , -2 , -3 , -4 , . . . .
We represent the negative integers on a line as being on the other side of 0 from the positive integers, like this:
The positive integers, negative integers, and zero all together are called the integers. Thus —9, 0, 10, —5 are all integers.
OPERATIONS ON WHOLE NUMBERS
Addition
The Core Secret: Alignment & Carrying
Always line up the numbers from the right side (the Units place). It helps to put the largest number on top. You add column by column, moving from right to left.
The Rule: If a column adds up to a 2-digit number (like 20 or 18), you write down the right digit and carry over the left digit to the next column on the left.
Step-by-Step Breakdown
Line them up like this:
Example 1.
Find out:
a.) 98 + 6734 + 348
Solution
Step-by-Step Breakdown
Line them up like this:
Column 1 (Units): .
Write down 0, carry 2 over to the Tens column.
Column 2 (Tens): .
Write down 8, carry 1 over to the Hundreds column.
Column 3 (Hundreds): .
Write down 1, carry 1 over to the Thousands column.
Column 4 (Thousands): .
Drop down the 7.
Final Answer: 7180
I.e, 6349 + 259 +7954
98
6734
+ 348
7180
b.)
6349
+ 259
79542
86150
Subtracting
The Core Secret: Alignment & Borrowing
Always line up the numbers from the right side. You subtract column by column, moving from right to left.
- Rule 1: If the top number is bigger, just subtract.
- Rule 2: If the top number is smaller, borrow "1" from the neighbor on the left to make your top number bigger.
Example
Find: 73469 – 8971
Step-by-Step Breakdown
- Column 1 (Units): (Easy, 9 is bigger).
- Column 2 (Tens): 6 is too small. Borrow 1 from the 4 next door. The 4 becomes 3, and the 6 becomes 16. Now: .
- Column 3 (Hundreds): We have is too small. Borrow 1 from the 3 next door. That 3 becomes 2, and our 3 becomes 13. Now: .
- Column 4 (Thousands): We have Borrow 1 from the 7. The 7 becomes 6, and the 2 becomes 12. Now: .
- Column 5 (Ten-Thousands): Only the 6 is left. Drop it down: .
Final Answer: 64498
Self-Check Trick: Add your answer () to the bottom number (). If it equals the top number (), you got it right!
73469
- 8971
64498
Multiplication
The product is the result of two or more numbers.
Example 2.
Work out: 469 x 63
Solution
469
X 63
-------------
1407
+28140
-------------
29547
Division
Division is a mathematical operation that means sharing or grouping a number into equal parts.
When one number is divided by another, the result is called the quotient.
- The dividend is the number being divided.
- The divisor is the number that divides the dividend.
- The quotient is the result you get after dividing.
- If something is left over, it is called the remainder.
Example 3 (without remainder):
20 ÷ 4 = 5
Dividend: 20 (the number being divided)
Divisor: 4 (the number dividing)
Quotient: 5 (the result of division)
Since 4 × 5 = 20, there is no remainder.
Example 4 (with remainder):
22 ÷ 5 = 4 remainder 2
22 is the dividend
5 is the divisor
4 is the quotient
2 is the remainder
Whole-Number Long Division
Long division is a step-by-step method used to divide large numbers.
It helps you find how many times one number (the divisor) can go into another number (the dividend).
When you divide, the result is called the quotient, and if something is left over, it is the remainder.
Example 5; Divide 275 ÷ 7
Solution
Follow the steps below;
- How many times does 7 go into 2? 0 → look at the first two digits: 27.
- How many times does 7 go into 27? 3 times because 3×7=21.
- Write 3 above the 7 (in the tens place). Subtract: 27−21=6
- Bring down the next digit (5) → we have 65.
- How many times does 7 go into 65? 9 times because 9×7=63
- Write 9 above the 5 (one place). Subtract: 65−63=2
- No more digits to bring down → remainder = 2.
Answer: quotient = 39, remainder = 2.
Mathematically
39
┌────
7 | 275
-21
----
65
-63
----
2 ← remainder
FRACTION;
Definition of Fraction: Fractions is a part of a whole, it is to represent the portion/part of the whole thing. It represents the equal part of the whole. A fraction has two parts, namely numerator and denominator. The number on the top is called the numerator and the bottom is called the denominator.
TYPES OF FRACTIONS:
1. Proper fraction
2. Improper fraction
3. Mixed fraction
- Proper Fraction: Proper fraction is when the numerator is smaller than the denominator.
Examples of proper fractions are 
, 
and so on.
- Improper Fraction: Improper fraction is when the numerator is smaller than the denominator.
Examples of improper fractions are 
and so on.
- Mixed Fraction: This is the combination of a whole number and a proper fraction.
Examples of Improper fractions are
i) 
, Where 4 is the whole number and 
is a proper fraction.
ii) 
, Where 6 is the whole number and 
is a proper fraction.
OPERATION IN FRACTIONS
Addition and Subtraction
The numerators of fractions whose denominators are equal can be added or subtracted directly.
Example 6
+ =
6/8 – 5/8 = 1/8
When adding or subtracting numbers with different denominators like:
5/4 + 3 /6=?
2/5 – 2/7 =?
Step 1– Find a common denominator (a number that both denominators will go into or L.C.M)
Step 2– Divide the denominator of each fraction by the common denominator or L.C.M and then multiply the answers by the numerator of each fraction
Step 3– Add or subtract the numerators as indicated by the operation sign
Step 4 – Change the answer to lowest terms
Example 7
Multiplying Simple Fractions
Step 1– Multiply the numerators
Step 2– Multiply the denominators
Step 3– Reduce the answer to lowest terms by dividing by common divisors
Example 8
Multiplying Mixed Numbers
Step 1– Convert the mixed numbers to improper fractions first
Step 2– Multiply the numerators
Step 3– Multiply the denominators Step 4– Reduce the answer to lowest terms
Example 9
Dividing Simple Fractions
Step 1– Change division sign to multiplication
Step 2– Change the fraction following the multiplication sign to its reciprocal (rotate the fraction around so the old denominator is the new numerator and the old numerator is the new denominator)
Step 3- Multiply the numerators
Step 4– Multiply the denominators Step 5– simplify the answer to lowest terms
Example 10
Dividing Mixed Numbers
Step 1 – Convert the mixed number or numbers to improper fraction.
Step 2 – Change the division sign to multiplication.
Step 3– Change the fraction following the multiplication sign to its reciprocal (flip the fraction around so the old denominator is the new numerator and the old numerator is the new denominator)
Step 4- Multiply the numerators.
Step 5– Multiply the denominators. Step 6– Simplify the answer to lowest form.
Example 11
EVEN NUMBER;
A number which can be divided by 2 without a remainder E.g. 0,2,4,6 0 or 8 3600, 7800, 806, 78346
ODD NUMBER:
Any number that when divided by 2 gives a remainder. E.g. 471,123, 1197,7129.The numbers end with the following digits 1, 3, 5,7 or 9.
DECIMAL POINT;
A decimal point can be defined as a point or dot which is used to separate a whole number from the fraction part of a number.
PRIME NUMBER;
Prime number is a natural number greater than one, which has only two factors i.e. ‘1’ and itself.
Examples are; 
NB; 1 is not a prime number and 2 is the only even number which is a prime number.
COMPOSITE NUMBER;
Composite is a natural or a positive integer that has more than two factors.
Examples are; 
and so on.
A FACTOR:
A factor of a number is a number that divides the given number evenly or exactly leaving no remainder.
NB: Since division by ‘0’ is undefined then ‘0’ can never be a factor.
Example;
DIVISION | REMAINDER | IS THE NUMBER A FACTOR |
| 4 R O | YES |
| 7 R 0 | YES |
| 5 R O | YES |
Divisibility Tests Table
The divisibility test makes computation of numbers easier. The following is a table for the divisibility test.
Divisibility Tests | Example |
A number is divisible by 2 if the last digit is 0, 2, 4, 6 or 8. | 168 is divisible by 2 since the last digit is 8. |
A number is divisible by 3 if the sum of the digits is divisible by 3. | 168 is divisible by 3 since the sum of the digits is 15 (1 + 6 + 8 = 15), and 15 is divisible by 3. |
A number is divisible by 4 if the number formed by the last two digits is divisible by 4. | 316 is divisible by 4 since 16 is divisible by 4. |
A number is divisible by 5 if the last digit is either 0 or 5. | 195 is divisible by 5 since the last digit is 5. |
A number is divisible by 6 if it is divisible by 2 and it is divisible by 3. | 168 is divisible by 6 since it is divisible by 2 and it is divisible by 3. |
A number is divisible by 8 if the number formed by the last three digits is divisible by 8. A number is divisible by 9 if the sum of the digits is divisible by 9. A number is divisible by 10 if the last digit is 0. A number is divisible by 11 if the sum of its digits in the odd positions like 1st ,3rd ,5th ,7th Positions, and the sum of its digits in the even position like 2nd , 4th ,6th ,8th pos | 7,120 is divisible by 8 since 120 is divisible by 8. 549 is divisible by 9 since the sum of the digits is 18 (5+4+9=18), and 18 is divisible by 9. 1,470 is divisible by 10 since the last digit is 0. 8,260,439 sum of 8 +6 +4 +9 =27: 2 + 0 +3 = 5 ; 27 – 5 = 22 which is a multiple of 11 |
DECIMAL PLACE VALUE;
This refers to the place values of all digits in a given decimal number which includes the whole number part and the fractional part. Let assume we have
from above the place value of each digit are;
1 – Thousands
2 – Hundreds
3 – Tens
4 – Unit
5 – Tenth
6 – Hundredth
7 – Thousandths
8 – 10 Thousandths
Shifting of Decimals when multiply by 
and so on.
OPERATION ON DECIMALS
Addition and Subtraction The key point with addition and subtraction is to line up the decimal points!
Example 12:
2.64 + 11.2
2.64
+11.20 (In this case, it helps to write 11.2 as 11.20 for alignment)
-----------
13.84
Example 13: 14.73 – 12.155
14.730 (Adding this 0 helps with alignment and calculation)
- 12.155
--------
2.575
Example 14: 127.5 + 0.127
127.500 (It helps to add zeros for consistent decimal places)
+ 0.127
----------
127.627
Multiplication when multiplying decimals, do the sum as if the decimal points were not there, and then calculate how many numbers were to the right of the decimal point in both the original numbers - next, place the decimal point in your answer so that there are this number of digits to the right of your decimal point?
Example 15; 2.1 x 1.2. Calculate 21 x 12 = 252. There is one number to the right of the decimal in each of the original numbers, making a total of two.
We therefore place our decimal so that there are two digits to the right of the decimal point in our answer.
Hence 2.1 x 1.2 =2.52. Always look at your answer to see if it is sensible. 2 x 1 = 2, so our answer should be close to 2 rather than 20 or 0.2 which could be the answers obtained by putting the decimal in the wrong place.
Example 16; 1.4 x 6
Calculate 14 x 6 = 84. There is one digit to the right of the decimal in our original numbers so our answer is 8.4
Check 1 x 6 = 6 so our answer should be closer to 6 than 60 or 0.6
When dividing decimals, the first step is to write your numbers as a fraction. Note that the symbol / is used to denote division in these notes.
Hence:
2.14 / 1.2 =
Next, move the decimal point to the right until both numbers are no longer decimals. Do this the same number of places on the top and bottom, putting in zeros as required.
Hence this can then be calculated as a normal division. Always check your answer from the original to make sure that things haven’t gone wrong along the way. You would expect to be somewhere between 1 and 2. In fact, the answer is 1.78.
Example 17; 4.36 / 0.14
Solution
= = in 2 decimal place
BODMAS RULE:
Bodmas is an acronym and it is an order you have to follow while solving mathematical expressions.
B - Bracket ()
O – Of
D – Division
M – Multiplication
A - Addition
S - Subtraction
Examples: Using BODMAS Rule simplify the following;
Example 18; 
Step-by-step solution:
- Brackets first →
- Expression becomes: 
- Division and Multiplication (from left to right)
- Expression becomes: 
- Subtraction
Example 19. 
PERFECT SQUARE
In mathematics are positive integers obtained by squaring an integer, meaning multiplying an integer by itself.
Example 20; 4 is a perfect square since it a product of an integer with itself i.e 
Also the product of 
PERFECT SQUARE ROOT
The perfect square root of a number is a positive integer that, when squared, results in a perfect square.
Example 21:
i) 
is a perfect square because 
ii) The perfect square root of 
RULES FOR ADDITION
Integers follow very simple rules for addition. These are:
Commutativity. If a, b are integers, then a + b = b + a. For instance, we have 3 + 5 = 5 + 3 = 8, or in an example with negative numbers, we have -2 + 5 = 3 = 5 + (-2 ).\
Associativity. If a, b, c are integers, then (a + b) + c = a + (b + c).
In view of this, it is unnecessary to use parentheses in such a simple context, and we write simply For instance, We write simply a+b+c
Example 22; For an instance;
(3 + 5) + 9 = 8 + 9 = 17,
3 + (5 + 9) = 3 + 14 = 17.
Associativity also holds with negative numbers. For example,
Example 23;
( -2 + 5) + 4 = 3 + 4 = 7,
- 2 + (5 + 4) = - 2 + 9 = 7.
Also
(2 + (-5 )) + (-3 ) = - 3 + (-3 ) = -6 ,
2 + ( - 5 + (-3 )) = 2 + (-8 ) = -6 .
BASIC RULES OF MULTIPLICATION AND DIVISION OF MATHEMATICAL SIGN/ OPERATION
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We can multiply integers, and the product of two integers is again an integer. We shall list the rules which apply to multiplication and to its relations with addition. We again have the rules of commutativity and associativity respectively:
and
We emphasize that these apply whether a, b, c are negative, positive, or zero.
Multiplication is also denoted by a dot. For instance
3 • 7 = 21, and
(3 • 7) • 4 = 21 • 4 = 84,
3 • (7 . 4) = 3 • 28 = 84.
For any integer a, the rules of multiplication by 1 and 0 are:
,
Example 24; We have
Distributive Rule of Multiplication over Addition and Subtraction
a{b + c) = ab + ac and also on the other side,
(b + c)a = ba + ca.
Also note that, . Thus .
Also Adding -a to both sides, we obtain
NB: - (3a) = ( -3)a = 3 (-a ).
4 (a — 5b) = 4a — 20b.
- 3 (5a - 7b) = -15a + 21b.
NB; ( — l)a = —a.
Proof. We have ( -1)a + a = ( -1)a + la = ( - 1 + 1)a = Oa = 0.
By definition, ( — l)a + a = 0 means that ( — l)a = —a, as was to be shown. We have
— (ab) = ( —a)b.
Also, we must show that ( — a)b is the negative of ab. This amounts to showing that
ab + ( —a)b = 0. But we have by distributivity ab + ( —a)b = (a + ( — a))b = 0b = 0, thus proving what we wanted.
NB; — (ab) = a( —b).
Example 22. We have .
Similarly, . Note that the product of two minus signs gives a plus sign.
Example 23; We have ( — 1) ( — 1) = I To see this, all we have to do is apply our rule
— (ab) = ( —a)b = a( — b).
We find ( - l ) ( - l ) = - ( l ( - 1 ) = - ( - l ) = l
Example 24. More generally, for any integers a, b
we have ( —a) ( —b) = ab.
Example. A product of a negative number and a positive number is negative.
For instance, —4 is negative, 7 is positive, and ( — 4) • 7 = - (4 . 7 ) = -28, so that (—4) • 7 is negative.
When we multiply a number with itself several times, it is convenient to use a notation to abbreviate this operation. Thus we write
and in general if n is a positive integer, (the product is taken n times).
We say that an is the n-th power of a. Thus, a² is the second power of a, and a⁵ is the fifth power of a.
If m and n are positive integers, then:
Example 25:
Example 26:
Example 27:
NB
LCM & HCF (Lowest Common Multiple & Highest Common Factor)
i) LCM (Least Common Multiple): The smallest multiple that two or more numbers share.
ii) HCF (Highest Common Factor): The largest number that can divide two or more numbers exactly.
Example 28: Find the LCM and HCF of 12 and 18.
Common factor |
|
|
2 | 12 | 18 |
3 | 6 | 9 |
2 | 2 | 3 |
3 | 1 | 3 |
| 1 | 1 |
- LCM(12,18) =
The LCM = 
HCF(12,18) =
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The H.C.F 
STANDARD FORM:
Standard form is a way of writing down very large or very small numbers easily.
Also, it is a way of writing a very large number with one number before the decimal point, multiplied by a power of 10, e.g. 580,000 can also be written as .
Example 29: Consider the number 
Step 1: Write the first number => 
Step 2: Add a decimal point after this and write the remaining non-zero numbers => 
Step 3: Now count the number of digits after =>
There are 7 digits. The 7 digits will be the power of 10 while writing the given number in standard form
Step 4: So in Standard form; 
Example 30: Simplify 
leaving your answer in standard form..
Solution
=> 
Example 31: Simplify 
leaving your answer in standard form.
Solution;
=> 
Example 32: Simplify 
leaving your answer in standard form.
Answer;
=> 
Reflection | The scope of this course is highly ambitious, covering foundational concepts (Integers, Basic Operations, Fractions, Factors) and advanced rules (BODMAS, Standard Form). We plan to ensure that students build strong foundational understanding while developing confidence in applying these concepts to multi-step problems,particularly involving integers, fractions and decimals. |
Class Quiz Questions |
Part 1: Whole Numbers Addition
Subtraction
Part 2: Decimals Addition
Subtraction
Part 3: Fractions Addition
Subtraction
2. Division Exercises (Using Long Division) Instructions: For each problem, perform a long division to find the quotient and remainder (if any). Divide 345 by 5 Divide 728 by 7 Divide 1,280 by 8 Divide 947 by 4 Divide 275 by 7 Divide 5,618 by 6 Divide 3,000 by 12 Divide 4,567 by 15 Decimal Division (Using Long Division) Instructions: Perform long division. Adjust the divisor to be a whole number first (move decimal points), then divide. Round your quotient to two decimal places if necessary.
3. Justify each step, using commutativity and associativity in proving the following identities.
4.
5. Solve for x in the following equations.
6. Prove the cancellation law for addition: If a + b = a + c, then b = c. 7 Prove: If a + b = a, then b=0. 8. Express each of the following expressions in the general form:9. Prove:First equation: Second equation: 10. Obtain expansions for:Obtain expansions for and similar to the expansions for and of the preceding exercise. Expand the following expressions as sums of powers of x multiplied by integers. First expression: Second expression: 11. Expand the following expressions as sums of powers of x multiplied by integers. 12.
13. i). Write 27707807 in words ii).All prime numbers less than ten are arranged in descending order to form a number a.) Write down the number formed b.) What is the total value of the second digit? c.) Write the number formed in words. 14. Simplify. 15. Express 1.93+0.25 as a single fraction 16. Simplify the following leaving your answer in standard form.
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