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:
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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