I won't presume to tell you whether or not the least significant parts of the mantissa are important for the work that you are doing. I will suggest that you can view those least significant digits in the HP-41 in the same way that you viewed the so-called "guard digits" in the TI and Casio machines. Some examples follow. Start with the calculator in Fix 9.

Playing with pi:

1. Place the value of pi in the display; i.e. the ten digit value 3.141592654

2. Divide pi by 4 and see 0.785398164 in the display but you know that the pi value in step 1 divided by 4 should be 0.7853981635 .

3. To see all ten digits of the mantissa simply multiply the displayed value (x register contents) by ten and see 0.7853981635

4. Place pi in the display and see 3.141592654

5. Square the displayed value and see 9.869604404

6. Square the displayed value again and see 97.40909109

7. Take the reciprocal of the displayed value and see 0.010265982; i.e., ten digts but only eight digits of the mantissa.

8. Multiply by 100 and see 1.026598225; i.e., all ten digits of te mantissa.

By now you should be able to see that what was done to see the least significant digits of the mantissa is the same kind of thing that you do with TI and Casio machines to see the guard digits.

Some trigonometry in DEG mode:

1. Enter 45, press sin and see 0.707106781 in the display.

2. Multiply by ten and see 7.071067812; i.e., all ten digits of the mantissa in the display.

3. Again, enter 45, press sin and see 0.707106781 in the display.

4. Enter 45, press cos and see 0.707106781 in the display.

5. Subtract and see 0.000000000 in the display.

Some trigonometry in RAD mode:

1. Enter pi, divide by 4, press sin and see 0.707106781 in the display.

2. Enter pi, divide by 4, press cos and see 0.707106781 in the display.

3. Subtract and see 2.0000000- 10 in the display.

To understand the result in step 3 above:

1. Enter pi, divide by 4, press sin and multiply by ten to see 7.071067813; i.e., all ten digits of the mantissa of the sine of pi/4.

2. Enter pi, divide by 4, press cos and multiply by ten to see 7.07106811; i.e., all ten digits of the mantissa of the cosine of pi/4.

3. Note that the two mantissas are different by two in the least significant place.

4. Why are the sine and cosine of pi/4 slightly different? Because the actual value of pi/4 is 0.78539816339... while the value of pi/4 in the HP-41 is 0.7853981635 slightly higher than the exact value. The sine of an angle slightly higher than pi/4 should yield a value slightly higher than the sine of pi/4. The cosine of an angle slightly higher than pi/4 should yield a value slightly lower than the cosine of pi/4. You can do the arithmetic.