# Decimal-Binary-Hex-Base7 Conversions with Radix Points

7segmentdisplaybinarycomputer-architectureconversionhex

I was hoping someone on here could help explain number base conversions with Radix points. I'm pursuing my Bachelors Degree in Computer Science with 12 courses remaining, currently in a Computer Hardware course.

For one of my assignments, I had to complete the following table (grey areas are the values I was given to convert to each of the others):

Here is more of my work for the other conversions not directly shown in the table:

``````DecToBin(37) = (37/2=1) (18/2=0) (9/2=1) (4/2=0) (2/2=0) (1/2=1)
=== 101001 or in reverse order 100101
DecToBin(99) = (99/2=1) (49/2=1) (24/2=0) (12/2=0) (6/2=0) (3/2=1) (1/2=1)
=== 1100011 or in reverse order 1100011
DecToBin(256) =  (598/2=0) (299/2=1) (149/2=1) (74/2=0) (37/2=1) (18/2=0) (9/2=1) (4/2=0) (2/2=0) (1/2=1) === 0110101001 or in reverse order 1001010110
DecToBin(3243) = (3243/2=1) (1621/2=1) (810/2=0) (405/2=1) (202/2=0) (101/2=1) (50/2=0) (25/2=1) (12/2=0) (6/2=0) (3/2=1) (1/2=1)  === 1101 0101 0011 or in reverse order 1100 1010 1011
DecToB7(37) = (37/7) = 5r2 or 52;
DecToB7(99) = (99/7)  = 14r1, (14/7) = 2r0; == 201;
DecToB7(170) = (170/7) = 24r2; (24/7) = 3r3; == 332;
DecToB7(1757) = (1757/7) = 251r0; (251/7) = 35r6; (35/7) = 5r0; == 5060;
DecToB7(598) = (598/7) = 95r3; (95/7) = 13r4; (13/7) = 1r6;
DecToB7(3243) = (3243/7) = 463r2; (463/7) = 66r1; (66/7) = 9r3; (9/7) = 1r2; == 12312;
B7toDec(12) = (1*7^1)+(2*7^0) = (7)+(2) = 9 in Decimal
B7toDec(666) = (6*7^2)+(6*7^1)+(6*7^0) = 342 in Decimal
DecToBin(9) = (9/2=1) (4/2=0) (2/2=0) (1/2=1) === 1001 or in reverse order 1001
DecToBin(342) = (342/2=0) (171/2=1) (85/2=1) (42/2=0) (21/2=1) (10/2=0) (5/2=1) (2/2=0) (1/2=1)
=== 011010101 or in reverse order 101010110
``````

Now I've been given a second table to do the same, but this one I am to include calculating all the values to four places after the radix point. My book itself isn't very clear only talking about radix points in a chapter for 2 sentences, and what I've found online hasn't really "clicked" for me.

Can someone take a stab at explaining how I would convert with the radix points as described? I'm not looking for answers, just for someone to provide some examples which will enable me to understand what is being asked and how to go about it.

Any and all assistance appreciated! This class has been tough all around thus far and I'm still playing catch up in a few areas đź™‚

EDIT:

Can anyone confirm if I'm on the right track going over to Decimal?

``````BinaryToDec(11011.0111) = 2^4+2^3+0+2^1+2^0 . +0+2^-2+2^-3+2^-4 = 30.4375
BinaryToDec(111.0111) = 2^2+2^1+2^0 . +2^-1+0+2^-3+2^-4 = 7.6875
HexToDec(2.08) = (2*16^0)+. (0)+(8*16^-2) = 2.03125
HexToDec(AB.C) = (10*16^1)+(11*16^0)+ . (12*16^-1) = 171.75
Base7ToDec(1.2) = (1*7^0) . + (2*7^-1) = 1.285714286
Base7ToDec(66.6) = (6*7^1)+(6*7^0)+ . (6*7^-1) = 48.85714286
``````

EDIT2:

Making some progress:

``````DecToBin(0.25) = 2^-2 = 0.25 or 0.0100
DecToBin(0.35)=
BinaryToDec(11011.0111) = (2^4)+(2^3)+(0)+(2^1)+(2^0) + (0)+(2^-2)+(2^-3)+(2^-4) = 27.4375
BinaryToDec(111.1011) = (2^2)+(2^1)+(2^0) + (2^-1)+(0)+(2^-3)+(2^-4) = 7.6875
HexToDec(2.08) = (2*16^0) + (0)+(8*16^-2) = 2.03125 or by specs 2.0313
HexToDec(AB.C) = (10*16^1)+(11*16^0) + (12*16^-1) = 171.75 or by specs 171.7500
Base7ToDec(1.2) = (1*7^0)+(2*7-1) = 1.285714286 or by specs 1.2857
Base7ToDec(66.6) = (6*7^1)+(6*7^0) + (6*7^-1) = 48.85714286 or by specs 48.8571
http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm
DecToBinary(0.25) = 0.25*2 = 0.50 (0.0???); 0.50*2 = 1.00 (0.01??); 00*2 = 0, so answer is 0.0100
DecToBinary(0.35) = 0.35*2 = 0.70 (0.0???); 0.70*2 = 1.4 (0.01??); 0.4*2 = 0.80 (0.010?);
0.80*2 = 1.60 (0.0101????); 0.6*2 = 1.20 (0.01011???); 0.2*2 = 0.4 (0.010110??);
0.4*2 = 0.8 (0.0101100?); 0.8*2 = 1.60 (0.01011001????);
0.60*2 = 1.20 (0.010110011???); 0.2*2 = 0.4 (0.0101100110??);
0.4*2 = 0.8 (0.01011001100?); 0.8*2 = 1.60 (0.010110011001?);
DecToBinary(2.0313) = 2.0313*2 = 4.0626 (1?.????); 0.626*2= 1.252 (10.????);
0.252*2 = 0.504; (10.0???); 0.504*2 = 1.008 (10.01??);
0.008*2 = 0.016 (10.010?); 0.016*2 = 0.032 (10.0100???);
0.032*2 = 0.064 (10.01000??); 0.064*2 = 0.128 (10.010000?);
0.128*2 = 0.256 (10.0100000???); 0.256*2 = 0.512 (10.01000000??);
0.512*2 = 1.024 (10.010000001???); or by specs 10.0100â€¦.
``````

EDIT3:

More progress. Still having some issues with Decimal->Base7 Conversions:

``````DecToBin(0.25) = 2^-2 = 0.25 or 0.0100
DecToBin(0.35)=
BinaryToDec(11011.0111) = (2^4)+(2^3)+(0)+(2^1)+(2^0) + (0)+(2^-2)+(2^-3)+(2^-4) = 27.4375
BinaryToDec(111.1011) = (2^2)+(2^1)+(2^0) + (2^-1)+(0)+(2^-3)+(2^-4) = 7.6875
HexToDec(2.08) = (2*16^0) + (0)+(8*16^-2) = 2.03125 or by specs 2.0313
HexToDec(AB.C) = (10*16^1)+(11*16^0) + (12*16^-1) = 171.75 or by specs 171.7500
Base7ToDec(1.2) = (1*7^0)+(2*7-1) = 1.285714286 or by specs 1.2857
Base7ToDec(66.6) = (6*7^1)+(6*7^0) + (6*7^-1) = 48.85714286 or by specs 48.8571

http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm

DecToBinary(0.25) = 0.25*2 = 0.50 (0.0???); 0.50*2 = 1.00 (0.01??); 00*2 = 0, so answer is 0.0100
DecToBinary(0.35) = 0.35*2 = 0.70 (0.0???); 0.70*2 = 1.4 (0.01??); 0.4*2 = 0.80 (0.010?);
0.80*2 = 1.60 (0.0101????); 0.6*2 = 1.20 (0.01011???); 0.2*2 = 0.4 (0.010110??);
0.4*2 = 0.8 (0.0101100?); 0.8*2 = 1.60 (0.01011001????);
0.60*2 = 1.20 (0.010110011???); 0.2*2 = 0.4 (0.0101100110??);
0.4*2 = 0.8 (0.01011001100?); 0.8*2 = 1.60 (0.010110011001?);
DecToBinary(2.0313) = 2.0313*2 = 4.0626 (1?.????); 0.626*2= 1.252 (10.????);
0.252*2 = 0.504; (10.0???); 0.504*2 = 1.008 (10.01??);
0.008*2 = 0.016 (10.010?); 0.016*2 = 0.032 (10.0100???);
0.032*2 = 0.064 (10.01000??); 0.064*2 = 0.128 (10.010000?);
0.128*2 = 0.256 (10.0100000???); 0.256*2 = 0.512 (10.01000000??);
0.512*2 = 1.024 (10.010000001???); or by specs 10.0100â€¦.
DecToBinary(171.7500) = (171) = 128+0+32+0+8+0+2+1 = 10101011.   0.7500*2 = 1.500 (1.1???);
0.500*2 = 1.000 (1.11??) === 10101011.1100;
DecToBinary(1.2857) = 1.2857*2 = 2.5714 (1.????); 0.5714*2 = 1.1428 (1.1???); 0.1428*2 = 0.2856 (1.10??); 0.2856*2 = 0.5712 (1.100?); 0.5712*2 = 1.1424 (1.1001); 0.1424*2 = 0.2848 (1.10010???); 0.2848*2 = 0.5696 (1.100100??); 0.5696*2 = 1.1392 (1.1001001??); 0.1392*2 = 0.2784 (1.10010010â€¦) or by specs 1.1001â€¦
DecToBinary(48.8571) = (48) = 00110000. 0.8571*2 = 1.7142 (0.1????); 0.7142*2 = 1.4284 (0.11???); 0.4284*2 = 0.8568 (0.110??); 0.8568*2 = 1.7136 (0.1101?); 0.7136*2 = 1.4272 (0.11011) ===
110000.11011 or by specs 110000.1101

DecToHex(0.25) = 0.??? = 0.25*16 = 4; (0.4??);
DecToHex(0.35) = 0.??? = 0.35*16 = 5.6 (0.5??); 0.6*16 = 9.6 (0.59???); 0.6*16 = 9.6 (0.599???); === 0.5999999999â€¦.. or by specs 0.5999
DecToHex(27.4375) = (27) = 1B; 0.4375*16 = 7; === 1B.7
DecToHex(7.6875) = (7) = 7; 0.6875*16 = 1.375 (7.1??); 0.375*16 = 6 (7.16); === 7.B
DecToHex(1.2857) = (1) = 1; 0.2857*16 = 0.5714 1.492
3A;
DecToHex(48.8571) =48/16 = 3?.????; 0/16 = 0 (30.????);  0.8571*16 = 13.7136 (30.D???); 0.7136*16 = 11.4176 (30.DB???); 0.4175*16 = 6.6816 (30.DB6??); 0.6816*16 = 10.9056 (30.DB6A?); 0.9056*16 = 14.4896 (30.DB6AE?); 0.4896*16 = 7.8336;

DecToBase7(0.25) = 0.??? = 0.25*7 = 1.75 (0.1??); 0.75*7 = 5.25 (0.15??); 0.25*7 = 1.75 (0.151?);
``````

EDIT4:

So, clearly I am doing something wrong with my Decimal to Base7 conversions. I finally found a calculator that accepts fractional decimals, but the only conversion I am coming up correctly with is the bottom row where the Base7 value was already provided:

Can anyone spot what I'm doing wrong in my Math?

``````DecToBase7(0.25) = (0*7^0)+(2*7^-1)+(5*7^-2) =  (0) + (0.2857142857) + (0.1020408163) = 0.387755102 or by specs 0.3877.
DecToBase7(0.35) = (0*7^0)+(3*7^-1)+(5*7^-2) = (0)+(0.4285714286) + (0.1020408163) = 0.5306122449 or by specs 0.5306.
DecToBase7(27.4375) = (2*7^1)+(7*7^0)+(4*7^-1)+(3*7^-2)+(7*7^-3)+(5*7^-4) = 21.65514369 or by specs 21.6551.
DecToBase7(7.6875) = (7*7^0)+(6*7^-1)+(8*7^-2)+(7*7^-3)+(5*7^-4) = 8.042898792 or by specs 8.0428
``````

EDIT5:

Thanks to everyone who helped me get a better grasp of Decimal/Binary/Hex/Base7 conversions with/without Radix points! I've got the table fully filled out and believe it's close if not all correct. I'll double check later, but for now here is the table and my calculations I used to arrive at the values (hopefully it'll help someone else down the road):

``````DecToBin(0.25) = 2^-2 = 0.25 or 0.0100
DecToBin(0.35)=
BinaryToDec(11011.0111) = (2^4)+(2^3)+(0)+(2^1)+(2^0) + (0)+(2^-2)+(2^-3)+(2^-4) = 27.4375
BinaryToDec(111.1011) = (2^2)+(2^1)+(2^0) + (2^-1)+(0)+(2^-3)+(2^-4) = 7.6875
HexToDec(2.08) = (2*16^0) + (0)+(8*16^-2) = 2.03125 or by specs 2.0313
HexToDec(AB.C) = (10*16^1)+(11*16^0) + (12*16^-1) = 171.75 or by specs 171.7500
Base7ToDec(1.2) = (1*7^0)+(2*7-1) = 1.285714286 or by specs 1.2857
Base7ToDec(66.6) = (6*7^1)+(6*7^0) + (6*7^-1) = 48.85714286 or by specs 48.8571
``````
``````DecToBinary(0.25) = 0.25*2 = 0.50 (0.0???); 0.50*2 = 1.00 (0.01??); 00*2 = 0, so answer is 0.0100
DecToBinary(0.35) = 0.35*2 = 0.70 (0.0???); 0.70*2 = 1.4 (0.01??); 0.4*2 = 0.80 (0.010?);
0.80*2 = 1.60 (0.0101????); 0.6*2 = 1.20 (0.01011???); 0.2*2 = 0.4 (0.010110??);
0.4*2 = 0.8 (0.0101100?); 0.8*2 = 1.60 (0.01011001????);
0.60*2 = 1.20 (0.010110011???); 0.2*2 = 0.4 (0.0101100110??);
0.4*2 = 0.8 (0.01011001100?); 0.8*2 = 1.60 (0.010110011001?);
DecToBinary(2.0313) = 2.0313*2 = 4.0626 (1?.????); 0.626*2= 1.252 (10.????);
0.252*2 = 0.504; (10.0???); 0.504*2 = 1.008 (10.01??);
0.008*2 = 0.016 (10.010?); 0.016*2 = 0.032 (10.0100???);
0.032*2 = 0.064 (10.01000??); 0.064*2 = 0.128 (10.010000?);
0.128*2 = 0.256 (10.0100000???); 0.256*2 = 0.512 (10.01000000??);
0.512*2 = 1.024 (10.010000001???); or by specs 10.0100â€¦.
DecToBinary(171.7500) = (171) = 128+0+32+0+8+0+2+1 = 10101011.   0.7500*2 = 1.500 (1.1???);
0.500*2 = 1.000 (1.11??) === 10101011.1100;
DecToBinary(1.2857) = 1.2857*2 = 2.5714 (1.????); 0.5714*2 = 1.1428 (1.1???); 0.1428*2 = 0.2856 (1.10??); 0.2856*2 = 0.5712 (1.100?); 0.5712*2 = 1.1424 (1.1001); 0.1424*2 = 0.2848 (1.10010???); 0.2848*2 = 0.5696 (1.100100??); 0.5696*2 = 1.1392 (1.1001001??); 0.1392*2 = 0.2784 (1.10010010â€¦) or by specs 1.1001â€¦
DecToBinary(48.8571) = (48) = 00110000. 0.8571*2 = 1.7142 (0.1????); 0.7142*2 = 1.4284 (0.11???); 0.4284*2 = 0.8568 (0.110??); 0.8568*2 = 1.7136 (0.1101?); 0.7136*2 = 1.4272 (0.11011) ===
110000.11011 or by specs 110000.1101
``````

``````DecToHex(0.25) = 0.??? = 0.25*16 = 4; (0.4??);
DecToHex(0.35) = 0.??? = 0.35*16 = 5.6 (0.5??); 0.6*16 = 9.6 (0.59???); 0.6*16 = 9.6 (0.599???); === 0.5999999999â€¦.. or by specs 0.5999
DecToHex(27.4375) = (27) = 1B; 0.4375*16 = 7; === 1B.7
DecToHex(7.6875) = (7) = 7; 0.6875*16 = 1.375 (7.1??); 0.375*16 = 6 (7.16); === 7.B
DecToHex(1.2857) = (1) = 1; 0.2857*16 = 0.5714 1.492
3A;
DecToHex(48.8571) =48/16 = 3?.????; 0/16 = 0 (30.????);  0.8571*16 = 13.7136 (30.D???); 0.7136*16 = 11.4176 (30.DB???); 0.4175*16 = 6.6816 (30.DB6??); 0.6816*16 = 10.9056 (30.DB6A?); 0.9056*16 = 14.4896 (30.DB6AE?); 0.4896*16 = 7.8336;
``````

To verify Base7 results: http://korn19.ch/coding/base_converter.php

``````DecToBase7(0.25) = 0.???? = 0.25*7 = 1.75 (0.1???); 0.75*7 = 5.25 (0.15??); 0.25*7 = 1.75 (0.151?); 0.75*7 = 5.25 (0.1515â€¦);
DecToBase7(0.35) = 0.???? = 0.35*7 = 2.45 (0.2???); 0.45*7 = 3.15 (0.23??); 0.15*7 = 1.05 (0.231?); 0.05*7 = 0.35 (0.2310);
DecToBase7(27.4375) = 36.???? = 0.4375*7 = 3.0625 (36.3???); 0.0625*7 = 0.4375 (36.30??); 0.4375*7 = 3.0625 (36.303?); 0.0625*7 = 0.4375 (36.3030);
DecToBase7(7.6875) = 10.???? = 0.6875*7 = 4.8125 (10.4???); 0.8125*7 = 5.6875 (10.45??); 0.6875*7 = 4.8125 (10.454?); 0.8125*7 = 5.6875 (10.4545);
DecToBase7(2.0313) = 2.???? = 0.0313*7 = 0.2191 (2.0???); 0.2191*7 = 1.5337 (2.01??); 0.5337*7 = 3.7359 (2.013?); 0.7359*7 = 5.1513 (2.0135);
DecToBase7(171.7500) = 333.???? = 0.7500*7 = 5.25 (333.5???); 0.25*7 = 1.75 (333.51??); 0.7500*7 = 5.25 (333.515?); 0.25*7 = 1.75 (333.5151);
DecToBase7(1.2857) = 1.???? = 0.2857*7 = 1.9999 (1.1???); 0.9999*7 = 6.9993 (1.16??); 0.9993*7 = 6.9951 (1.166?); 0.9951*7 = 6.9657 (1.1666);
``````

I think I spotted it:

DecToBase7(0.25) = (0*7^0)+(2*7^-1)+(5*7^-2) = (0) + (0.2857142857) + (0.1020408163) = 0.387755102 or by specs 0.3877.

Yes, that is how I convert a 0.25_(base 7) to a base-ten decimal value.

To convert 8.8750_(decimal) to a base-7 value, you need to go the other way:

``````8.8750_(decimal)                            1
1
.
0.8750_(decimal) * 7 = 6.1250_(decimal) --> 6
0.1250_(decimal) * 7 = 0.8750_(decimal) --> 0
0.8750_(decimal) * 7 = 6.1250_(decimal) --> 6
0.1250_(decimal) * 7 = 0.8750_(decimal) --> 0
0.8750_(decimal) * 7 = 6.1250_(decimal) --> 6
0.1250_(decimal) * 7 = 0.8750_(decimal) --> 0
0.8750_(decimal) * 7 = 6.1250_(decimal) --> 6
...                               ...
``````

So 8.8750_(decimal) is equal to 11.60606060606...(base 7).

With a few special exceptions, converting a fraction in any one base to a fraction in any other base usually leads to such repeating patterns. (Perhaps the most notorious such unexpected repeating pattern happens when converting 0.1_(decimal) to binary. A similar repeating pattern occurs when converting 0.1_(base 3) to decimal).