Basic Concepts of Algebra

1. NUMBER

 
1.  NUMBER

1.1              PLACE VALUE and DEFINITIONS:
Our number system is based on ten, the number of fingers and thumbs we have. This number system is called the base ten system / the denary system / the decimal system. This system uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The place of a digit in a number tells you the value of that digit.
                        Let us consider the number 7,954 the value of the digit 4 is 4 units, the value of the digit 5 is 5 tens, the value of the digit 9 is 9 hundreds and the value of the digit 7 is 7 thousands.
                        Let us consider the number 502. Here, zero is used to show that the tens column is empty. So the denomination is 5 hundreds, o tens and 2 units. Extra columns may be added to represent larger numbers, where the value of each column is 10 times greater than the column on its immediate right.
                        The numbers 0 to 9, all inclusive can be represented, utilized and manipulated to be renamed resulting in a special definition. Some of the definitions are:

Whole numbers - These numbers are zero and the counting numbers, example:  0, 1, 2, 3, 4...

Natural / counting numbers – These numbers are whole numbers greater than zero example:1, 2, 3, 4, 5 ...
Negative numbers – These numbers are less than zero, example: . . .-5,-4,-3,-2,-1(Also  -1/10,-1/100, or fraction of the kind)
Positive numbers – These numbers are whole numbers and fractions greater than zero, example: 0,1/12,1/4,8/11,23/17,4, ...
Fractions –These numbers may lie anywhere on the number line between two whole numbers, between -1 and zero or between negative numbers, example: -5/4,-1/7,-3/4,21/5…)
Integers –These numbers are negative counting numbers and whole numbers, example: … -3, -2, -1, 0, 1, 2, 3 …
Even numbers –These are numbers into which 2 divides exactly, example: 2, 4, 6, 8, 10…

Odd numbers –These are numbers into which 2 does not divide exactly, example: 1, 3, 5, 7, 9, 11, …

Square number –This is the result of a rational number multiplied by itself. It is sometimes called a perfect square or simply a square, example: 1, 4, 9, 16, 25… i.e. 1'1=1, 2'2=4, 3'3=9, . . . -1*-1=1,-2*-2=4,-3*-3=9,…
Triangle number –This is a natural number that may be represented as a triangle of dots. Each triangle number is obtained from the previous one by adding one more row of dots @ a base to extend the diagonal, example: 1, 3, 6, 10, 15, 21 … 

R1                    .              .                              .                                            .
R2                                    .      .                      .      .                                    .      .                                   
R3                                                                  .      .      .                            .      .      .     
R4                                                                                                                .      .      .      .

Multiple –When two or more numbers are multiplied the result is called a multiple.


Factor –A number that can divide exactly into a multiple is a factor. Example 2'5'4=40
40 is the multiple; 2, 4, 5 are factors.
Prime number –This is a number which has two factors, itself and 1, example: 2, 3, 5, 7, 11 . . .               

Cube number –This is the resulting multiple when a factor appears in its product thrice, otherwise expressed as a factor raised to a power/index of three (3). The symbol '^' will be used to mean 'to the power of' (exponent).
Example: 1'1'1 = 1^3=1...i.e. 1 x 1x 1 = 1 to the power of 3=1.
                  -1'-1'-1 = (-1)^3=-1
                  2'2'2 = 2^3=8
                  -2'-2'-2 = (-2)^3=-8
Composite number –This is a number which has more than two factors, example: 4, 6, 9, 12, 14 . . .

Rational numbers –These numbers can be expressed in the form of a/(b≠0). These numbers include all the integers and positive and negative fractions, example: -3, -2.8, √4, 11/8, 7. . .
Irrational numbers –These are numbers that cannot be expressed as the ratio of two integers, example: π,√(2 ),...
Real numbers –These numbers are the set of rational and irrational numbers.

Imaginary numbers –These numbers are represented as the root of a negative number, example: √(-1), √(-3). . . These numbers are written in the form  bi,ci,di,. . .
Complex numbers –These numbers are expressed in the form  a+bi, where a is called the real part and bi is called the imaginary part, example: 4+3i where i is the number √(-1) .


Diagram 1.


Numbers:
*Complex  --Real & Imaginary
*Real  --Rational & Irrational
*Rational --Fractions & Integers
*Integers –Negative Integers, Zero & Positive Integers.



  EXERCISE 1.1

1.  Write these numbers in figures.
    (a) Twenty seven      (b) seventy    (c) one hundred and five    (d) fifteen thousand
    (e) seven hundred and twelve thousand.

2.  What is the value of the 2 in each of following numbers?
    (a) 142    (b) 3,254    (c) 52,375    (d) 200,005

3.  Write the numbers in the question 2 above in words.

4.  Write down the largest number and the smallest number you can make with the digits below:
    (a) 5, 0, 7 and 3    (b) 9, 2, 3 and 5

5.  List all the numbers in each case:
    (a) Even numbers between 17 and 27    (b) Odd numbers between 10 and 24
    (c) The first 5 square numbers

6.  List all the numbers of the following set.
    (a) Integers greater than -2 but less than 5    (b) Multiples of 7 that are less than 27
    (c) A composite number that is a factor of 10

7.  Write down the prime factors of 60.

8.  Round off 75,483 to the nearest hundred.

9.  What is the greatest common factor of 12 and 20?

10. Find the sums of the first two, three, four, five cube numbers. What do you notice about your   
      Answer?




1.2 ARITHMETIC OPERATIONS:
                        There are four common operations of arithmetic, namely: addition, subtraction, multiplication and division. An operation works to change numbers and so we can create special operations for specific uses. One such operation is the exponent (power). It is wise to note that raising a number to a fractional power is the same as taking a root of that number; example: x^1/3 is the same as ∛x
Each of the operations has a special symbol and a name for the result thus: 4+5 = 9← (sum) result
                                                                                                                          ↑plus sign/symbol
Addition and subtraction are reverse processes:
                        Thus: 9 - 5      =    4
                                      ↑              ↑   
                                  Minus    Difference
               
            (Minuend) (Operation sign) (Subtrahend) (Difference)
                    ↓                        ↓                      ↓ 
                    9                      -                    5            =        4

                          Multiply
                              ↓
Multiplication:      4' 5 = 20 ← result
                              ↑
                          Product of 4 by 5

Division:    20 ' 5 = 4 ← quotient
                      ↑
                      divide

Exponent:  2^3= 2 ' 2 ' 2 = 8← result...... Note: The symbol '^' means to the power of.
                    8^1/3 = ∛8 = 2^1    ← result.






    EXERCISE 1.2
1.  Add these numbers:

(a)  15                    (b) 15                    (c)  840
        22                              24                          259
        31                          31                          100
   

2.  Perform these subtractions:

(a)  112                    (b) 121                  (c) 6432
      -  13                          -18                        -759

3. Multiply these numbers:

(a)  18                      (b)  180                (c) 987
    ' 14                            '  49                    ' 347

4. Perform these division problems:

(a) 4  148            (b) 32  6489    (c)  72          856

5. Given that a3 = * a* a ;  √a = a 1/2
    Calculate the values of:

(a) 8^3                        (b) 9^3                  (c) 9^1/2 





1.3 NUMBER PROPERTIES

Identity –This is a special kind of number. For example, zero is called an additive identity because when you add zero to a number, the number remains the same.
0 + 2 = 2  ;    0 + 3 =3  ;  0 + x=x ;  0 + y=y
                  The number 1 is called a multiplicative identity because when you multiply a number by 1 , the number remains the same.
10 ' 1 = 10  ;  12 ' 1 = 12 ;    x ' 1 = x ;  y ' 1 = y

Inverse –A number’s additive inverse is a number you can add to the original number  to get the additive identity (zero) For example, the additive inverse of 15 is -15, because 15 + -15 = 0
                  A number’s multiplicative inverse is a number you can multiply by the original number to get the multiplicative identity (one). For example, the multiplicative inverse of 15 is 1/15  because 15 x 1/15 = 1, likewise 1/15 ' 15 = 1.

Associative  property – this is an operation whereby you can group numbers in any way without changing the answer. Addition and multiplication are both associative. For example:
2 + (3 + 4) = (2 + 3) + 4 = 3+ (2 + 4)    ;    2' (3 ' 4) = (2 ' 3) '4 = 3' (2 ' 4)

NOTE: Subtraction is NOT associative:
(4 - 3) -2 ≠ 4 – (3 – 2) and neither is division: (4 ' 3) ' 2 ≠ 4 ' (3 ' 2)

Commutative property –This is an operation whereby you can change the order of the numbers without changing the result. Addition and multiplication are both commutative,
For example: 17 + 12 = 12 + 17             
                        ⇒  29      = 29
                 
7 ' 12 = 12 '7
⇒ 84      =  84
NOTE: subtraction is NOT commutative:
12 – 7 ≠ 7 – 12 and neither is division
12/7      ≠ 7/12

Distributive property –This is the act of giving out things. Thus, if a teacher gave word cards to 5 boys and 4 girls so that each received 2 cards. What was the total numbers of cards given to the 9 students?          
Solution:
                5 + 4 = 9 students (boys and girls)
                          X 2 words cards each
                          18 word cards all together
or distributed as:
                              2 ' (5+4) = 2 '5 + 2 '4
                              2 ' 9            =    10    +    8
                                            18    =    18          

NOTE: Multiplication is distributive over addition and subtraction but NOT distributive over division
-        Exponents cannot be distributed over addition or subtraction.
-        Remember that EXPONENT is distributive over multiplication.

(3 '2)^3  =  3^3  ' 2^3
      6^3        =  27 ' 8
      216    =  216

   




      EXERCISE 1.3
      Calculate the following:

1.  514 + 86 + 0

2.  100 + 200 + 0

3.  225 ' 0

4.  18 '0 ' 20

5.  75+ (-75)

6.  15 '  1/15

7.  (8 + 12) + 10

8.    9 + (12 – 10)

9.    8 + 12 + x = 12 + 10 + 8

10. (3 ' 2)^3  = 3^3' 2^x
      ∴ x = ?
 





1.4    ORDER OF OPERATIONS:
                              It is wise to note that the order in which the arithmetic operations is carried out will affect the result of the math task. A convention is designed to be used so as to avoid confusion.
                              The order in which arithmetic operations must be carried out is as follows:
Brackets (parentheses) first, then Divide, then Multiply, then Add, then Subtract. (BODMAS)
Example:
1.4a. Work out the following:  (5 + 6) ' 3 – 1

SOLUTION: (5 + 6) ' 3 – 1
                      5 + 6 = 11
                      11'3 = 33
                      33 -  1 = 32
BODMAS: The expression has Brackets, Multiplication, and Subtraction. So do B, then M and then S.
1.4b.          10 + (8 + 4) ' 3      Use B, then D, then A.
                    →  (8 + 4) = 12
                          12 ' 3 = 4
                          10 + 4 = 14.

       




  EXERCISE 1.4
  Calculate the following:

1.  (3 ' 4) + 12 ' 4

2.  (12 ' 6) ' 2

3.  24 ' (4 + 2) -1

4.  15 ' (7 -2) + 6

5.  4 ' 8 + 3
         





1.5  SIGNED or DIRECTED NUMBERS:
                        When the temperature reading on the thermometer is -5*F we say that this is cold. A temperature reading of 0*F is also cold but 70*F is warm. These numbers; -5*, 0* and +70*F are signed numbers -5 means 5* below 0 and 70* means 70* above 0. ( *F means degrees F).
                        A good way to understand signed numbers is to use number line to represent a thermometer placed in a horizontal position.
                   
                          -15  -10  -5  0  5  10  15  20  25  30  35  40

The numbers to the left of zero are shown with the (-) sign while the numbers to the right of zero are understood to be plus(+).  Zero (0) has no sign, neither positive nor negative.

Definitions:
                  A signed/directed number is a number preceded by a plus or minus sign written to the left of a number in a superscript position; for example: 5^(0 ) below zero is written-5^o. The operations: Addition(+), Subtraction (-), Multiplication (') and Division (') are written on the line of the directed numbers. Thus, +5 + -2, has 5 in the positive direction, plus (operation) 2 in the negative direction. The  absolute value of a signed number is the number that remains when the sign is removed. For example, the absolute value of -5 is 5 likewise the absolute value of +70 is 70. Instead of writing the words absolute value we can use the symbol / /. Thus, /-5/ is read the absolute value of negative 5.
Thus /-5/ = 5
And /+ 70/ = 70

                Operation with signed /directed numbers
1.5A.  - ADDITION OF SIGNED NUMBERS>
RULE 1:  To add signed numbers with LIKE SIGNS, add their absolute values, then prefix the common sign.

Example 1.5A1
                (+4) + (+3) = +7    (+ + + = +)
                (-4) + (-3) = -7      (- + - = -)
RULE 2: To add two numbers with unlike signs, subtract the smaller absolute value from the larger, then prefix the sign the number having the larger absolute value.

Example 1.5A2
                (-4) + (+3) = (-1)
NOTE: /-4/ = 4, and /+3/ = 3. Subtract the smaller absolute value, 3, from the larger absolute value,4, then prefix the sign of the larger absolute value 4. Thus, (-4) + (+3) = (-1).



           


EXERCISE 1.5A                      (DO NOT USE A CALCULATOR)

1. (^+ 7 + (^+ 5) =

2. (^- 7) + (^- 5) =

3. (^+ 7) + (^+ 5) + (^+ 3) =

4. (^- 8) + (^- 7) + (^- 6) =

5. (^- 10) + (^+ 2) =

6. (^+ 7) + (^- 5) =

7. (^- 1) + (^+ 4) + (^+ 2) =

8. (^+ 9) + (^- 2) + (^- 3) =

9. (^- 3) + (^+ 6) + (^- 2) + (^+ 5)=

10. (^+ 3) + (^- 6) + (^+ 2) + (^- 5)=






1.5B: –SUBTRACTION OF SIGNED NUMBERS.
RULE 3: To subtract signed numbers you change the sign of the number to be subtracted (second number) and add.
EXAMPLE 1.5B1
                  (^+ 7) – (^- 2)
NOTE:      Change the sign of the negative 2, it becomes positive 2, (+2) ; Then add  +2 to +7 (i.e. subtraction sign/minus becomes addition sign/plus).
⇒ (^+ 7) – (^- 2) = (^+ 7) + (^+ 2) = (^+ 9)
EXAMPLE:  1.5B2
                  (^- 7) – (^+ 2)
⇒              (^- 7)- (^+ 2) = -7 + -2 = (^- 9)







EXERCISE 1.5B                    (DO NOT USE A CALCULATOR) 

1.  25- 18 =         

2.  14- 24 =

3.  (^+ 12) - (^- 10)=

4.  (^+ 18)-(14) =

5.  (^- 19) - (^- 8)=

6.  -7-4=

7.  -15 – 7 =

8.  (^+ 5)- (^- 5)=

9.  (^- 11) – (^+ 4)=

10. (99)- (^- 99)=






1.5C – MULTIPICATION OF SIGNED NUMBERS:
RULE 4. To multiply two signed numbers with like signs, multiply their absolute values and make the product positive.
EXAMPLE: 1.5C1.
                  (^+ 3) ' (^+ 4) =^+ 12
                  (^- 3) ' (^- 4) =^+ 12

RULE 5. To multiply two signed numbers with unlike signs, multiply their absolute values and make the product negative.

Example 1.5C2
                      (^- 3) ' (^+ 4) =^- 12
                (^+ 3) ' (^- 4) =^- 12







EXERCISE 1.5C                    (DO NOT USE A CALCULATOR)

1.  (^+ 5) ' (^+ 3)=

2.  (^- 7) ' (^- 2)=

3.  4 '5 =

4.  (^- 5) (^- 2)=

5.  (^+ 3) (^- 5)=

6.  (^- 6) (^+ 4)=

7.  (^- 4) ' (^- 2)=

8.  5 (^- 4)=

9.  (^- 7) (^+ 8)=

10.(^- 1) (^- 1)=
                         






RULE 6.  To multiply an even number of negatives, multiply their absolute value and make the product positive.
Example 1.5C_3
                        -5 ' -4 ' -3 X −2  i.e. four signed numbers, four is an even number
                    → 5 ' 4 ' 3'2 = (+120)
RULE 7.  To multiply an odd number of negatives, multiply their absolute values and make the product negative.

EXAMPLE: 1.5C4
                          -5 ' -4 ' -3  i.e. three signed numbers, three is an odd number
                    → -5 ' -4 '-3 = -60
Now try these problems:
1.  (^- 2) ' (^- 3) ' (^- 5) ' (^- 10)=
2.  (^- 2) ' (^- 3) ' (^- 5) =
3.  (^- 3) '(^- 5) ' (^- 7) ' (^- 2) ' (^- 1) ' (^- 10)=
4.  (^+ 1) ' (^+ 2) ' (^+ 3) ' (^+ 4) ' (^+ 5)'(+10)=
5.  (^+ 1) ' (^+ 2) ' (^+ 3) ' (^+ 4) ' (^+ 5)=

1.5D – DIVISION OF SIGNED NUMBERS:
RULE8.  To divide two signed numbers with like signs, divide their absolute values and make the quotient positive.
EXAMPLE: 1.5D1
                            (^+ 10) ' (^+ 2) = (^+ 5)
                            (^- 10) ' (^- 2) = (^+ 5)
RULE9.  To divide two signed numbers with unlike signs, divide their absolute values and make the quotient negative.
EXAMPLE: 1.5D2
                            (^+ 10) ' (^- 2) = (^- 5)
                            (^- 10) ' (^+ 2) = (^- 5)







      EXERCISE 1.5D                    (DO NOT USE A CALCULATOR)

1.  24 ' 18

2.  (^+ 14) ' (^+ 2)

3. (^- 12) ' (^- 4)

4.  (^- 18) ' (^+ 3)

5.  (^+ 40) ' (^- 5)