Number Bases are the combination of digits that a system of counting use to represent Numbers. If you have been finding it so difficult to count in number bases, worry no more because we have gotten easier ways when dealing with NUMBER SYSTEM.

Number system is a unique way of writing to express numbers.

**Let us take a good look at the bases given below.**

## Number Bases

**BASE “Y”** **INTEGER**

Base Two 0 1

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”.

**We can have:**

745_{eight}

745_{nine}

745_{ten}

**But we can not have:**

745_{seven}, 745_{six}, 745_{five}.

**If the integer is 214, we can have:**

214_{five}

214_{six}

214_{seven} till base ten.

Have you asked yourself the reason why the bases written above started 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 and not in figures.*

E.g. 214_{five} and 745_{seven} etc. Instead of 214_{5} and 745_{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.

### Knowing The **P**lace** V**alue** 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 _{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_{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 (Number Bases**)**:**

**Write down the following in the powers of their number bases.**

**Q1. 3412**_{five}

_{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**_{two}

_{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**_{six}

_{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**_{nine}

_{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**_{three}

_{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**_{nine}

_{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**_{four}

_{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**_{seven}

_{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**_{two}

_{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**_{three}

_{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°

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