Sure, let's solve the given problems step-by-step.
### 1) Writing the decimal number in short form.
The expanded form of the decimal number given is:
[tex]\[ 2 \times 10^2 + 4 \times 10^1 + 3 \times 10^0 \][/tex]
We will evaluate each term separately:
[tex]\[ 2 \times 10^2 = 2 \times 100 = 200 \][/tex]
[tex]\[ 4 \times 10^1 = 4 \times 10 = 40 \][/tex]
[tex]\[ 3 \times 10^0 = 3 \times 1 = 3 \][/tex]
Now, add these values together:
[tex]\[ 200 + 40 + 3 = 243 \][/tex]
So, the decimal number in short form is:
[tex]\[ 243 \][/tex]
### 2) Converting the decimal number into binary.
Next, we need to convert the decimal number [tex]\( 243 \)[/tex] into a binary number.
The binary number system is a base-2 numeral system which uses only two digits, [tex]\( 0 \)[/tex] and [tex]\( 1 \)[/tex].
To convert a decimal number to binary, we repeatedly divide the number by 2 and record the remainders.
Let's convert [tex]\( 243 \)[/tex]:
[tex]\[ 243 \div 2 = 121 \quad \text{remainder} \ 1 \][/tex]
[tex]\[ 121 \div 2 = 60 \quad \text{remainder} \ 1 \][/tex]
[tex]\[ 60 \div 2 = 30 \quad \text{remainder} \ 0 \][/tex]
[tex]\[ 30 \div 2 = 15 \quad \text{remainder} \ 0 \][/tex]
[tex]\[ 15 \div 2 = 7 \quad \text{remainder} \ 1 \][/tex]
[tex]\[ 7 \div 2 = 3 \quad \text{remainder} \ 1 \][/tex]
[tex]\[ 3 \div 2 = 1 \quad \text{remainder} \ 1 \][/tex]
[tex]\[ 1 \div 2 = 0 \quad \text{remainder} \ 1 \][/tex]
Reading the remainders from bottom to top, we get:
[tex]\[ 11110011 \][/tex]
Thus, the binary representation of the decimal number [tex]\( 243 \)[/tex] is:
[tex]\[ 11110011 \][/tex]
So, the solutions are:
1. The decimal number in short form is [tex]\( 243 \)[/tex].
2. The binary representation of [tex]\( 243 \)[/tex] is [tex]\( 11110011 \)[/tex].
