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If you have been finding it so difficult to count in number bases, worry no more because we have gotten an easier way when dealing with NUMBER SYSTEM.
Let us take a good look at the bases given below.
Base Three 0 1 2
Base Four 0 1 2 3
Base Five 0 1 2 3 4
Base Six 0 1 2 3 4 5
Base seven 0 1 2 3 4 5 6
Base Eight 0 1 2 3 4 5 6 7
Base Nine 0 1 2 3 4 5 6 7 8
Base Ten 0 1 2 3 4 5 6 7 8 9
You can see from the illustration above that for every number base Y, the highest digit possible in the whole number/integer given must be at least 1 lesser than the number base "Y".
If for instance we are given an integer "745", you can see in the integer we have 7, 4 and 5 as its digits.
So, the number base it must have should be "greater" than the "highest" digit in the integer "745".
745 base Eight
745 base Nine
745 base Ten.
But we can not have
745 base seven
745 base six
745 base five and so on.
If the integer is 214, we can have:
214 base six
214 base seven till base ten.
Have you asked yourself why the bases only start from FIVE?. This is because in the integer 214, the highest digit there is 4 and we have been told that the number base MUST be at least 1 greater than the highest digit in the given integer.
Do not be confused on these two because they are the same:
"The highest digit possible in the whole number/integer given must be at least 1 lesser than the number base" and "The number base must at least be 1 greater than the highest digit of the given integer".
One must have been wondering why we keep on writing the bases in words instead of digits.
E.g. 214 base five and 745 base seven etc. Instead of 214 base 5 and 745 base 7 respectively.
It doesn't call for any alarm, it is being written in words simply to avoid confusion between the integer and its base.
NOTE: Counting is done generally in base ten. Do not panic if you see when they use the words Decimal or Denary in number system because they both mean "Base ten" and base two is called Binary.
PLACE VALUE OF DIGITS IN INTEGERS.
Knowing the placement of each digit in number system is something you are supposed to take seriously serious.
Note that any positive integer raised to power zero is one(1), that is 0° = 1, 1° = 1, 2° = 1, 3° = 1, 4° = 1, 12° = 1 and so on.
For an integer "3964 base ten", the place value is 3 thousands, 9 hundreds, 6 tens, 4 units.
= 3 x 1000 + 9 x 100 + 6 x 10 + 4 x 1
= 3 x 10³ + 9 x 10² + 6 x 10¹ + 4 x 10°
For an integer "526 base eight", the place value is
5 Sixty fours, 2 eights, 6 units.
= 5 x 64 + 2 x 8 + 6 x 1.
= 5 x 8² + 2 x 8¹ + 6 x 8°
Also, for "1732 base nine".
= 1 seven hundred and twenty nine, 7 eighty one, 3 nine, 2 units.
= 1 x 729 + 7 x 81 + 3 x 9 + 2 x 1.
= 1 x 9³ + 7 x 9² + 3 x 9¹ + 2 x 9°
QUESTIONS AND SOLUTIONS.
Write down the following in the powers of their number bases.
Q1. 3412 base five
= 3 one hundred and twenty five, 4 twenty five, 1 five, 2 units.
= 3 x 125 + 4 x 25 + 1 x 5 + 2 x 1.
= 3 x 5³ + 4 x 5² + 1 x 5¹ + 2 x 5°
Q2. 11001 base two
= 1 sixteen, 1 eight, 0 four, 0 two, 1 units.
= 1 x 16 + 1 x 8 + 0 x 4 + 0 x 2 + 1 x 1.
= 1 x 2⁴ + 1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2°
Q3. 32004 base six
= 3 one thousand two hundred and ninety six, 2 two hundred and sixteen, 0 thirty six, 0 six, 4 units.
= 3 x 1296 + 2 x 216 + 0 x 36 + 0 x 6 + 4 x 1.
= 3 x 6⁴ + 2 x 6³ + 0 x 6² + 0 x 6¹ + 4 x 6°
Q4. 56021 base nine
= 5 six thousand five hundred and Sixty one, 6 seven hundred and twenty nine, 0 eighty one, 2 nine, one units.
= 5 x 6561 + 6 x 729 + 0 x 81 + 2 x 9 + 1 x 1.
= 5 x 9⁴ + 6 x 9³ + 0 x 9² + 2 x 9¹ + 2 x 9°
Q5. 2102 base three
= 2 twenty seven, 1 nine, 0 three, 2 units.
= 2 x 27 + 1 x 9 + 0 x 3 + 2 x 1.
= 2 x 3³ + 1 x 3² + 0 x 3¹ + 2 x 3°
Q6. 81250 base nine
= 8 six thousand five hundred and Sixty one, 1 seven hundred and twenty nine, 2 eighty one, 5 nine, 0 units.
= 8 x 6561 + 1 x 729 + 2 x 81 + 5 x 9 + 0 x 1.
= 8 x 9⁴ + 1 x 9³ + 2 x 9² + 5 x 9¹ + 0 x 9°
Q7. 33102 base four
= 3 two hundred and fifty six, 3 sixty four, 1 sixteen, 0 four, 2 units.
= 3 x 256 + 3 x 64 + 1 x 16 + 0 x 4 + 2 x 1.
= 3 x 4⁴ + 3 x 4³ + 1 x 4² + 0 x 4¹ + 2 x 4°
Q8. 26043 base seven
= 2 two thousand four hundred and one, 6 three hundred and forty three, 0 forty nine, 4 seven, 3 units.
= 2 x 2401 + 6 x 343 + 0 x 49 + 4 x 7 + 3 x 1.
= 2 x 7⁴ + 6 x 7³ + 0 x 7² + 4 x 7¹ + 3 x 7°
Q9. 110110 base two
= 1 thirty two, 1 sixteen, 0 eight, 1 four, 1 two, 0 units.
= 1 x 32 + 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 0 x 1.
= 1 x 2^5 + 1 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 0 x 2°
Q10. 10221 base three
= 1 eighty one, 0 twenty seven, 2 nine, 2 three, 1 units.
= 1 x 81 + 0 x 27 + 2 x 9 + 2 x 3 + 1 x 1.
= 1 x 3⁴ + 0 x 3³ + 2 x 3² + 2 x 3¹ + 1 x 3°