Converting bases to base ten (How to)

converting bases to base Ten
How to convert other bases to base ten

When converting bases to base ten, it has to undergo either of these two methods; a successive multiplication method or by power Expansion method. This simply means the reverse of the very one used when converting from base ten to another bases.

Method 1 (Converting bases): Successive multiplication.

This method of converting bases is the reverse of the consecutive division done when converting from base ten to another bases.

Method 2 (Converting bases): Power Expansion.

converting bases to base ten: This is where the use of place values comes in. We multiply out the number base using their place values.

Example 1:

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Convert 101101 base two to base ten.

Using consecutive/successive multiplication method for Converting bases.

converting bases to base ten:

Solution;

Integer given = 101101

Base given = base two

Base required = base ten

First step is to multiply the given base by the first digit (1)01101 of the integer given and adding the product to the nearest digit 1(0)1101.

So, 2 x 1 + 0 = 2 + 0 = 2

Second step is to multiply the result gotten from step one by the base given and add the third digit 10(1)101 of the integer to the product gotten.

2 x 2 + 1 = 4 + 1 = 5

Third step is to multiply the result gotten from the second step by the base given and add the forth digit 101(1)01 of the integer to the product gotten.

2 x 5 + 1 = 10 + 1 = 11

Forth step is to multiply the result gotten from third step by the base given and add the fifth digit 1011(0)1 of the integer to the product gotten.

2 x 11 + 0 = 22 + 0 = 22

Fifth step is to multiply the result gotten from forth step by the base given and add the final digit 10110(1) of the integer to the product gotten.

2 x 22 + 1 = 44 + 1 = 45 base ten.

Note the number of digits that make up the integer determine the number of steps to be taken.

Using Power Expansion to solve example 1.

101101 base two

= 1 x 2^5 + 0 x 2 ⁴ + 1 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2°

= 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1

= 32 + 0 + 8 + 4 + 0 + 1 = 45 base ten.

Students must be wondering how we got all those raised to powers.

Taking good look at this 101101

We devised this formula Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴…   to help us know the power that will be raised to the base at every digit of the integer.

Where

D = Digit

b = base

n = Total number of the digits.

Example 2.

Change 825 base nine to base ten.

Using successive multiplication method of converting bases to base ten.

Solution:

825 base nine

First step:

9 x 8 + 2 = 74

Second step:

9 x 74 + 5 = 671 base ten.

Using Power Expansion:

Solution:

825 base nine

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 8 x 9³–¹ + 2 x 9³–² + 5 x 9³–³

= 8 x 9² + 2 x 9¹ + 5 x 9° = 8 x 81 + 2 x 9 + 5 x 1

= 648 + 18 + 5  = 671 base ten.

QUESTIONS AND SOLUTIONS

Converting Bases: Change the following to base ten.

Q1. 132 base four

Solution

Method 1.

4 x 1 + 3 = 7

4 x 7 + 2 = 30 base ten

Method 2.

132 base four

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 1 x 4² + 3 x 4¹ + 2 x 4°

= 16 + 12 + 2  = 30 base ten.

Q2. 1026 base seven

Method 1

7 x 1 + 0 = 7

7 x 7 + 2 = 51

7 x 51 + 6 = 363 base ten

Method 2.

1026 base seven

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 7³ + 0 x 7² + 2 x 7¹ + 6 x 7°

= 343 + 0 + 14 + 6 = 363 base ten.

Q3. 49 base twelve

Method 1.

12 x 4 + 9 = 57 base ten

Method 2.

49 base twelve

= Dbⁿ–¹ + Dbⁿ–² = 4 x 12¹ + 9 x 12°

= 48 + 9 = 57 base ten.

Q4. 324 base six

Method 1.

6 x 3 + 2 = 20

6 x 20 + 4 = 124 base ten.

Method 2.

324 base six

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 3 x 6² + 2 x 6¹ + 4 x 6°

= 108 + 12 + 4 = 124 base ten.

Q5. 110110 base two

Method 1.

2 x 1 + 1 = 3

2 x 3 + 0 = 6

2 x 6 + 1 = 13

2 x 13 + 1 = 27

2 x 27 + 0 = 54 base ten.

Method 2.

110110 base two

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ + Dbⁿ–5 + Dbⁿ–6

= 1 x 2^5 + 1 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 0 x 2°

= 32 + 16 + 0 + 4 + 2 + 0 = 54 base ten.

Q6. 3120 base five

Method 1.

5 x 3 + 1 = 16

5 x 16 + 2 = 82

5 x 82 + 0 = 410 base ten.

Method 2.

3120 base five

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 3 x 5³ + 1 x 5² + 2 x 5¹ + 0 x 5°

= 375 + 25 +10 + 0 = 410 base ten.

Q7. 1000 base two.

Method 1.

2 x 1 + 0 = 2

2 x 2 + 0 = 4

2 x 4 + 0 = 8 base ten.

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 2³ + 0 x 2² + 0 x 2¹ + 0 x 2°

= 8 + 0 + 0 + 0 = 8 base ten.

Q8. 34 base six.

Method 1.

6 x 3 + 4 = 22 base ten.

Method 2.

= Dbⁿ–¹ + Dbⁿ–² = 3 x 6¹ + 4 x 6°

= 18 + 4 = 22 base ten.

Q9. 112 base three.

Method 1.

3 x 1 + 1 = 4

3 x 4 + 2 = 14 base ten.

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 1 x 3² + 1 x 3¹ + 2 x 3°

= 9 + 3 + 2 = 14 base ten.

Q10. 163 base eight.

Method 1.

8 x 1 + 6 = 14

8 x 14 + 3 = 115 base ten.

Method 2

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 1 x 8² + 6 x 8¹ + 3 x 8°

= 64 + 46 + 3 = 115 base ten.

Q11. 214 base nine

Method 1

9 x 2 + 1 = 19

9 x 19 + 4 = 175 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 2 x 9² + 1 x 9¹ + 4 x 9°

= 162 + 9 + 4 = 175 base ten

Q12. 1407 base eight

Method 1.

8 x 1 + 4 = 12

8 x 12 + 0 = 96

8 x 96 + 7 = 775 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 8³ + 4 x 8² + 0 x 8¹ +  7 x 8°

= 512 + 256 + 0 + 7 = 775 base ten

Q13. 230 base five

Method 1

5 x 2 + 3 = 13

5 x 13 + 0 = 65 base ten

Method 2

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 2 x 5² + 3 x 5¹ + 0 x 5°

= 50 + 15 + 0 = 65 base ten

Q14. 1104 base six

Method 1

6 x 1 + 1 = 7

6 x 7 + 0 = 42

6 x 42 + 4 = 256 base ten

Method 2

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴

= 1 x 6³ + 1 x 6² + 0 x 6¹ + 4 x 6°

= 216 + 36 + 0 + 4 = 256 base ten

Q15. 101101 base two

Method 1

2 x 1 + 0 = 2

2 x 2 + 1 = 5

2 x 5 + 1 = 11

2 x 11 + 0 = 22

2 x 22 + 1 = 45 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ + Dbⁿ–5 + Dbⁿ–6

= 1 x 2^5 + 0 x 2⁴ + 1 x 2³ + 1 x 2² + 0 x 2¹ + 1 x 2°

= 32 + 0 + 8 + 4 + 0 + 1 = 45 base ten

Q16. 10011 base two

Method 1.

2 x 1 + 0 = 2

2 x 2 + 0 = 4 

2 x 4 + 1 = 9

2 x 9 + 1 = 19 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ + Dbⁿ–5

= 1 x 2⁴ + 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2°

= 16 + 0 + 0 + 2 + 1 = 19 base ten

Q17. 2130 base four

Method 1

4 x 2 + 1 = 9

4 x 9 + 3 = 39

4 x 39 + 0 = 156 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ 

= 2 x 4³ + 1 x 4² + 3 x 4¹ + 0 x 4°

= 128 + 16 + 12 + 0 = 156 base ten

Q18. 56 base seven.

Method 1

7 x 5 + 6 = 41 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² 

= 5 x 7¹ + 6 x 7°

= 35 + 6

= 41 base ten

Q19. 841 base twelve

Method 1

12 x 8 + 4 = 100

12 x 100 + 1 = 1201 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³

= 8 x 12²  + 4 x 12¹ + 1 x 12°

= 1152 + 48 + 1 = 1201 base ten

Q20. 3241 base six

Method 1

6 x 3 + 2 = 20

6 x 20 + 4 = 124

6 x 124 + 1 = 745 base ten

Method 2.

= Dbⁿ–¹ + Dbⁿ–² + Dbⁿ–³ + Dbⁿ–⁴ 

= 3 x 6³ + 2 x 6² + 4 x 6¹ + 1 x 6°

= 648 + 72 + 24 + 1 = 745 base ten.

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