Question 1.1: Explore the addition and multiplication in the decimal, bina...
Explore the addition and multiplication in the decimal, binary, octal and hexadecimal number systems with some examples.
Decimal Arithmetic (For base 10, digits are 0 … 9)
\begin{aligned} &(1295)_{10}+(357)_{10}=(1652)_{10} \\ &(734)_{10} \times(46)_{10}=(33764)_{10} \end{aligned}
Binary Arithmetic (For base 2, digits are 0 and 1)
\begin{aligned} &(101011)_2+(11011)_2=(1000110)_2 \\ &(11101)_2 \times(1001)_2=(100000101)_2 \end{aligned}
Octal Arithmetic (For base 8, digits are 0 … 7)
\begin{aligned} &(1635)_8+(274)_8=(2131)_8 \\ &(752)_8 \times(23)_8=(22136)_8 \end{aligned}
Hexadecimal Arithmetic (For base 16, digits are 0 … 9, A, B, C, D, E, F)
\begin{aligned} &(5 AB 7)_{16}+( F 63)_{16}=(6 A 1 A )_{16} \\ &( A 4 B )_{16} \times(7 A )_{16}=(4 E 7 BE )_{16} \end{aligned}
Note: Arithmetic for numbers with base β:
Consider the addition of two numbers (1635)_8 \text { and }(274)_8 in the octal number system with the base β = 8. Note that, the addition of numbers 5 and 4 will produce number 9. For β = 8, we have 1 carry, and the remaining number is 1. Similarly, other calculations give the following result
\begin{aligned} &1 1 1 \text { Carry } \\ &(1635)_8 \\ &+(274)_8 \\\hline & (2131)_8 \\ &\Rightarrow(1635)_8+(274)_8=(2131)_8 \\ & \end{aligned}
Similarly, consider the multiplication of two numbers. For example, multiplication of numbers 7 and 5 will produce number 35. In octal system (base β = 8), for number 32, we have 4 carry; and remaining is 3. So, final result is (7)_8 \times(5)_8=(43)_8 .