Number base system of counting (Lesson 1)
The standard system of counting is in base 10 which is known as denary number. A denary number contains any of the digits 0,1,2,3,4,5,6, and 9. e.g 425 is a denary number, why? When, the number base is not specified, it is automatically base ten number (denary number). It can also be written as 42510.
Note: it contains the digits 4, 2 and 5 is among the possible digits of denary number.
Also, people count in following number bases:
a) base two known as binary number which contain only 0 and 1 e.g. 100012
b) base three known as ternary number which contain 0, 1, and 2 only e.g 21123
c) base four known as quaternary number which contain 0, 1, 2 and 3 only e.g. 3024
d) base five known as quinary number which contain 0, 1, 2, 3 and 4 only e.g. 24315
e) base 6 known as senary number which contain 0, 1, 2, 3, 4 and 5 only e.g. 53426
f) base 7 known as septenary number which contain 0, 1, 2, 3, 4, 5 and 6 only e.g. 64517
g) base 8 known as octal number which contain 0, 1, 2, 3, 4, 5,6 and 7 only e.g. 476428
h) base 9 known as nonary number which contain 0, 1, 2, 3, 4, 5, 6, 7 and 8 only e.g. 78459
i) base 10 known as denary number which contain 0, 1, 2, 3, 4, 5, 6, 7 8 and 9 only e.g. 6845910
j) base 11 known as undecimal number which contain 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and A e.g. 9A84511
k) base 12 known as duodecimal number which contain 0, 1, 2, 3, 4, 5, 6, 7 8, 9, A and B only e.g. 4AB612
l) base 13 known as tridecimal number which contain 0, 1, 2, 3, 4, 5, 6, 7 8, 9, A, B and C only e.g.
24C8B13
m) base 14 known as tetradecimal number which contain 0, 1, 2, 3, 4, 5, 6, 7 8, 9, A, B, C and D only e.g.
ABCD14
n) base 15 known as pentadecimal number which contain 0, 1, 2, 3, 4, 5, 6, 7 8, 9, A, B, C,D and E only
e.g. 3BCE515
o) base 16 known as hexadecimal number which contain 0, 1, 2, 3, 4, 5, 6, 7 8, 9, A, B, C,D and E only
e.g. 3BCE516
NOTE:
Any given type of number base can contain any of the digits less than its base number. For instance, a base nine number (nonary) must NOT have digit “9” as one of its digits.
Also, for number base above base 10 (denary), the each component of the number base may be 10, 11 …they denoted by letter as shown in the table below:
A B C D E F
10 11 12 13 14 15
Conversion of Number bases
1) Conversion of other bases to base ten (denary number)
In this lesson, the term “other bases” refer to any type of number base aside from denary number
(base ten). The method of converting other base number to base ten is called Expansion method.
Method of expansion
Step 1 - identify the number base of the given question
Step 2 - number each component of the question starting from the unit end (i.e. last number by
your extreme right) with 0, 1, 2... to the last number at your extreme left
Step 3 - multiply each component of the question by the base number raise to power of the
number on that component.
Step 4 - Add up the individual result obtained from step 3 to obtain your final answer
Example 1
i. Convert the following number bases to base ten
a) 10012 (b) 2123 (c) 845739 (d) 8A9F16
Solution
3 2 1 0
10012 = (1 x 23) + (0 x 22) + (0 x 21)+ (1 x 20) = (1 x 8) + (0 x 4) + (0 x 2) + (1 x 1) = 8 + 0 + 0 + 1 = 910
(23 = 2 x 2 x 2 =8, 20 = 1; Zero law of indices)
2 1 0
2123 = (2 x 32)+ (1 x 31)+ (2 x 30)= (2 x 9)+ (1 x 3)+ (2 x 1) = 18 + 3 + 2 = 2310
4 3 210
845739 = (8 x 94)+ (4 x 93)+ (5 x 92) + (7 x 91) + (3 x 90)= (8 x 6 561) + (4 x 729) + (5 x 81) + (7 x 9) + (3 x 1) = 52 488 + 2 916 + 405 + 63 +3 = 55 87510
3 2 1 0
8A9F16 = (8 x 163)+ (10 x 162) + (9 x 161) + (15 x 160)= (8 x 4096)+ (10 x 256) + (9 x 16) + (15 x 1)
= 32768 + 2560 + 144 + 15 = 35, 48710
ii. Arrange the following numbers in descending order of magnitude: 229 , 345 ,216 (WAEC)
Solution
To arrange in descending means to arrange the number from the highest to the lowest. However, there is a need to convert the base numbers given to base 10 for easier comparison
1 0
229 = (2 x 91) + (2 x 90)= (2 x 9)+ (2 x 1) = 18 + 2 = 2010
1 0
345= (3 x 51) + (4 x 50) = (3 x 5)+ (4 x 1) = 15 + 4 = 1910
1 0
216 = (2 x 61) + (1 x 60) = (2 x 6)+ (1 x 6 ) = 12 + 6 = 1810
Therefore, 229 > 345 > 216
Exercise
Convert the following number base to denary number:
(a)100012 (b) 64517 (c) 3024 (d) 21123 e. 4B3E15
Watch our youtube tutorial for the solutions and further explanation via: https://youtu.be/5RE-4rZPOkU
Conversion of number bases leading to equations
a) Solving number bases that lead to linear equations
Example 2
1) Find r, if 6 r 78 =5 1 19
2) Find n, if 34n= 100112
3) If 10112 + X7 = 2510
Solution
1) 6 r 78 =5 1 19
Using the expansion method to convert both LHS (base 8) and RHS (base 9) of the number bases equation given to base 10,
2 1 0 2 1 0
6 r 78 =5 1 19
(6 x82) + (r x81) + (7 x80) = (5 x92) + (1 x 91) + (1 x 90)
(6 x 64) + (r x8) + (7 x 1) = (5 x 81) + (1 x 9) + (1 x 1)
384 + 8r + 7 = 405 + 9 + 1
Re-arrange the LHS
384+7 +8r = 415
391+8r = 415 (Linear equation)
Collect the like terms by taking 391 to the RHS to become -391
8r = 415-391
8r = 24
Divide both sides by 8
8r/8 = 24/8
r=3
Note that LHS Left hand side of the equation, RHS –Right hand side of the equation)
2) 34n= 100112
Using the expansion method to convert both LHS (base n) and RHS (base 2) of the number bases equation given to base 10,
0 1 4 3 2 1 0
3 4n= 1 0 0 1 12
(3 x n1) + (4 x n0) = (1 x24) + (0 x 23) + (0 x 22) + (1 x 21) + (1 x 20)
(3 x n) + (4 x 1) = (1 x 16) + (0 x 8) + (0 x 4) + (1 x 2) + (1 x 1)
3n + 4 = 16 + 0 + 0 + 2 + 1
3n + 4 = 19
Collect the like terms by taking 4 to the RHS to become -4
3n = 19 – 4
3n = 15
Divide both sides by 3
3n/3 = 15/3
n=5
3) 10112 + X7 = 2510
Using the expansion method to convert both LHS (base 2) and (base 7) of the number bases equation given to base 10. RHS is base 10, so conversion is not necessary
3 2 1 0 0 0
1 0 1 12 + X7 = 2 510
(1 x 23) + (0 x 22) + (1 x 21) + (1 x 20) + (x 20) + (X x 70) = 25
(1 x 8) + (0 x 4) + (1 x 2) + (1 x 1) + (X x 1) = 25
8 + 0 + 2 + 1 + X = 25
11 + X = 25
Collect the like terms by taking 11 to the RHS to become -11
X= 25 – 11
X = 14
Exercise 2
1) If 1P034= 11510, find P
2) If 2q35 = 778, find q
