Base Numbers: Level 1

Base numbers: 

Convert other bases to base 10. 
Example #1:  
 1 | 0 |0₂ 
2²|2¹ |2⁰                  0*2⁰ + 0*2¹ + 1*2² = 4

Thus, 100₂  = 4 in base 10
The first few base 2 (or binary system) numbers that are equivalent to their base 10 (or decimal) counterpart.

base 2                       base 10

0                               0

1                               1

10                             2

11                             3

100                           4

101                           5

110                           6

111                           7

1000                         8

1001                         9

1010                        10    

Example #2:

1|  2|  4₅    

5²|5¹|5⁰                    4*5⁰ + 2*5¹ + 1*5²  = 4 + 10 + 25 = 39 

Thus, 124₅ = 39 in base 10   

Some important points: 
* All the digits in the number has to be smaller than the base. (Why?)
* Hexadecimal is base 16. In that system, you can use numbers from "0" to "9", the same as base 10.
  However, numbers 10 to 15, you use letters: A for 10, B for 11, C for 12, D for 13, E for 14 and last, F      for 15. 

Short cut to some base conversions:
 #1: Convert 1101111₂ to base 8.
Since 8 = 2³, you can convert base 2 to base 8 by regrouping every 3 digits.
Thus the original number will become 157₈
because if you regroup the original number starting from the right, you have 1₂-101₂-111₂ and it's 157 in base 2's term.
#2: Try convert 11110011₂ to base 4, which is 2² so you regroup by 2.

11 (3 in base 2)-11(3 in base 2)-00 (0 in base 2)-11(3 in base 2) so the answer is 3303_4. 

Convert a number from base 10 to any other base

Practice Problems: (answers below)

Convert other bases to base 10.

#: 10111₂                                           

#2_:  330₄

#3_:  100₅ 
#4_:  210₃

#5_:  24₆

#6_: Convert 10111₂  to base 4 

#7_: Convert 123321₄  to base 16 (or hexadecimal) [student Andrew's problem]

#8_:  11010011₂  is what to base 16 _?

#9_:  Convert 164₈ to binary (or base 2) [Daniel's question]

#10: What is 10000₅  minus 1_{5 ?} [Remaining :Willie's questions]

#11_: What is 0.1₈ to base 10?

#12_: What is 537₈  divided by 10_{8 ?}

#13_: 37₁₆  is what in base 4? what in base 10?

_#14_: What is 1231_{4  +} 2232₄ ?

_{Here is a link to help you practice more on level 1 base conversions.} 

_{Answer key:} 

#1: 23

#2: 60

#3: 25

#4: 21

#5: 16
#6: 23 4
#7: 6F9 16
#8: D3 16
#9: 1110100 2
#10: 1111 5
#11: 1/ 8
#12: 53.7 8
#13: 313 in base 4 and 55 in base 10
#14: 10123 4