Exploring Binary and Decimal Systems: Solving 112 in Different Bases

In numeral systems, the way we denote and manipulate numbers can vary widely. This article delves into the differences between the decimal (base 10) and binary (base 2) systems, with a special focus on understanding how 11 can equal 2 in a decimal system but 10 (which is 2 in decimal) in a binary system.

Understanding Different Bases

Numeral systems, or bases, are methods used to denote numbers. The most common base is the decimal system (base 10) which uses the digits 0 through 9. However, there are numerous other bases, each with its unique characteristics, such as the binary system (base 2), which uses just two digits, 0 and 1.

The Decimal System (Base 10)

Let's first review the decimal system. Here are the values for the first few powers of 10:

Units: 1 (100) Tens: 10 (101) Hundreds: 100 (102) Thousands: 1000 (103)

For example, the number 11 in the decimal system represents one ten (1×101) and one unit (1×100). Thus, 11 10 1 2 9 11.

The Binary System (Base 2)

Now, let's explore the binary system (base 2). The powers of 2 are:

Units: 1 (20) Two's place: 2 (21) Four's place: 4 (22) Eight's place: 8 (23) And so on...

In binary, the digits are 0 and 1. When you want to represent the number 2 in binary, you use the combination 10, which means 1 × 2 0 × 1 2. Similarly, 11 in binary is 3, as it is 1 × 2 1 × 1 2 1 3.

If 112 in Decimal, What Does 11 Equal in Binary?

Given that 11 2 in a decimal system, the key question is how to represent this number in binary. It's important to clarify that the notation is simply changing the base, not the value. When you express 11 in decimal as 2 9, in binary, it should be 1011, as follows:

1110  1×23   0×22   1×21   1×20

Breaking it down further:

23 (8) * 1 8 22 (4) * 0 0 21 (2) * 1 2 20 (1) * 1 1

Adding these together, 8 2 1 11, still in decimal.

However, to express 11 in binary, you simply convert the decimal value. 11 in binary is 1011, meaning 1×23 0×22 1×21 1×20.

Generalizing to Higher Bases

Understanding binary, let's generalize to higher bases. For any base n, the number 11 would be:

In decimal (base 10), 11 2 9. In binary (base 2), 11 1011 (8 2 1). In base 3, 11 10 (3 0).

The general formula for converting a number 11 into a different base n is to sum the products of each digit and the respective power of n. For example, if 11 is represented as 11 in some base, converting it to decimal involves the formula: 1×n 1.

Conclusion

While the numbers 11 and 2 can be confusing in various bases, understanding the positional notation in different numeral systems clears up these misunderstandings. Whether it is 11 in decimal being a specific value or 11 in binary being expressed as 1011, accurately interpreting the base is crucial. As you explore further into numeral systems, the flexibility and power of representation in different bases will become clearer.

By mastering the fundamentals of different bases, you can enhance your computational and problem-solving skills, making important contributions in various fields, from digital electronics to cryptography.