Converting and Adding Numbers in Different Bases: 121base4 84base16

Introduction to Base Systems

Understanding different base systems is crucial for computer science, cryptography, and various other fields. This article will guide you through the process of converting and adding numbers in base4, base16, and base10. Specifically, we'll work through an example problem: converting and adding 121 (base4) and 84 (base16).

Conversion to Base10

The first step in adding numbers in different bases is to convert them to base10, also known as the decimal number system. This allows us to perform arithmetic operations uniformly.

Conversion of 121(base4) to base10

Let's break down the conversion of 121(base4) to base10 step-by-step:

First, identify the place values in the base4 number. In base4, the rightmost digit is in the 40 (ones) place, the next digit to the left is in the 41 ( fours) place, and so on. Now, apply the formula for each digit's place value:

121base4 1 × 42 2 × 41 1 × 40

1. 42 16

2. 2 × 41 8

3. 1 × 40 1

Add these values together:

16 8 1 25

Therefore, 121base4 25base10

Conversion of 84(base16) to base10

Next, let's convert 84 (base16) to base10:

The rightmost digit is in the 160 (ones) place, and the next digit to the left is in the 161 (sixteens) place.

84base16 8 × 161 4 × 160

1. 8 × 161 128

2. 4 × 160 4

Add these values together:

128 4 132

Therefore, 84base16 132base10

Adding in Base10

Now that we have both numbers in base10, we can perform the addition:

25 (121base4) 132 (84base16) 157base10

Converting the Sum Back to Different Bases

Let's convert 157 (base10) back into base4 and base16.

Conversion of 157(base10) to base4

To convert 157 (base10) to base4, we divide the number by 4 repeatedly and keep track of the remainders:

157 ÷ 4 39 remainder 1 39 ÷ 4 9 remainder 3 9 ÷ 4 2 remainder 1 2 ÷ 4 0 remainder 2

Reading the remainders from bottom to top, we get 2131base4.

Conversion of 157(base10) to base16

To convert 157 (base10) to base16, we divide the number by 16 repeatedly and keep track of the remainders:

157 ÷ 16 9 remainder 13 (13 in hexadecimal is D) 9 ÷ 16 0 remainder 9

Reading the remainders from bottom to top, we get 9Dbase16.

Conclusion

In summary, converting and adding numbers in different bases involves several steps. First, we convert numbers to base10 for ease of arithmetic operation. After performing the addition, we can then convert the result back to the desired base. For the specific example of 121(base4) 84(base16), the final answer is:

121base4 84base16 25base10 132base10 157base10

157base10 2131base4

157base10 9Dbase16

Understanding these conversions and operations is essential for anyone dealing with number systems in computer science or mathematics.