The document discusses different numerical systems including: - Decimal - Uses base 10 and includes numbers like 1, 13, 2028, 12.1, 3.14159. - Binary - Uses the digits 0 and 1 and is the system used for computer processing. - Octal - A base-8 system that uses digits 0-7. - Hexadecimal - A base-16 system that uses digits 0-9 and letters A-F. It then explains different methods for converting between these systems including long division, using powers of the base, and a doubling method. Carried and borrowed digits are also discussed for addition and subtraction in these alternate bases.

1. Online class
Class 5
Prepared by:
Jeevan Subedi

2. The base-10 notational system for representing real numbers.
The expression of a number using the decimal system is called
its decimal expansion, examples of which include 1, 13, 2028,
12.1,
and 3.14159.
Binary
The system of representing text or computer processor
instructions by the use of the binary number system's two-binary
digits "0" and "1".
Octal
The base-8 number system, and uses the digits 0 to 7.
Numerals can be made from binary numerals by grouping
consecutive binary digits into groups of three (starting from the
right).
Hexadecimal
A positional numeral system with a radix or base of 16.

3.  Step 1—Divide the base 10 number by the radix (2) of t
h
e
binary system and extract the remainder (this becomes
the binary number's LSD).
 Step 2—continue the division by dividing the quotient o
f
step 1 by the radix (2
 Step 3—Continue dividing quotients by the radix until the
quotient becomes smaller that the divisor; then do one
more division. The remainder is our MSD.
The remainder in step 1 is our LSD. Now rewrite the
solution, and you will see that 510 equal 1012. Now follow
the conversion of 2310 to binary:
 Step 1—Set up the problem for division.
 Step 2—Divide the number and extract the remainder.
 Step 3—Rewrite the solution from MSD to LSD.

4. To convert a base 2 numbers to base 10, you
must know the decimal equivalent of each power
of 2.
The decimal value of a power of 2 is obtained by
multiplying 2 by itself the number of times
indicated by the exponent for whole numbers; for
example, 24 = 2 ´ 2 ´ 2 ´ 2 or 1610.
For fractional numbers, the decimal value is equal
to 1 divided by 2 multiplied by itself the number
of times indicated by the exponent.

5. This method does not use powers.
Start with the left-most digit of the given binary number.
For each digit as you move to the right, double your
previous total and add the current digit.
Note the first zero (italicized) is an initial subtotal for this
conversion process.
Like the positional notation method, this method can be
modified to convert from any base to decimal. Doubling is
used because the given number is of base 2. If the given
number is of a different base, replace the 2 in the method
with the base of the given number. For example, if the
given number is in base 37, you would replace the *2 with
*37. The final result will always be in decimal (base 10)

8. Carried Digits
+
1 1 1 1
1 1 0 1
1 0 1 1 1
----------------
1 0 0 1 0 0

9. Borrowed
-
* * * *
1 1 0 1 1 1 0
1 0 1 1 1
--------------------
1 0 1 0 1 1 1