We’ve talked about the **introduction and application of number bases **in one of our previous articles. If you’ve not seen it, you can easily watch the video below to better understand what number bases is all about.

Before we dive into the conversion process, let’s review some basic concepts. In any number base, each digit has a place value based on its position in the number. For example, in the decimal system, the rightmost digit represents ones, the next digit to the left represents tens, the next digit represents hundreds, and so on. The same idea applies to other number bases, except that the place values depend on the base itself. In the binary system, for example, the rightmost digit represents 2^{0} (which is 1), the next digit represents 2^{1} (which is 2), the next digit represents 2^{2} (which is 4), and so on.

To convert a number from another base to base 10, we need to multiply each digit by its corresponding place value and add up the results. Here’s a **step-by-step process for converting a number from another base to base 10**:

Step 1: Write down the number in expanded form Write the number in expanded form, with each digit multiplied by its place value. For example, if we want to convert the binary number 1101 to base 10, we would write:

(1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0)Step 2: Evaluate each term Evaluate each term using the exponentiation rule for the base. For example, in the above expression, we have:

1 \times 2^3 = 8, 1 \times 2^2 = 4, 0 \times 2^1 = 0, 1 \times 2^0 = 1Step 3: Add up the results Add up the results from Step 2 to get the decimal equivalent of the number. For example, in the above expression, we have:

8 + 4 + 0 + 1 = 13Therefore, the binary number 1101 is equal to the decimal number 13.

Let’s look at another example. Suppose we want to convert the octal number 74 i.e. 74_8 to base 10. We would write:

(7 \times 8^1) + (4 \times 8^0)Then we would evaluate each term:

7 \times 8^1 = 56, 4 \times 8^0 = 4And add up the results:

56 + 4 = 60Therefore, the octal number **74** is equal to the decimal number **60**.

Now let’s consider a more complex example. Suppose we want to convert the hexadecimal number **ABCD** to base 10. We would write:

Then we would evaluate each term:

A \times 16^3 = 10 \times 16^3 = 409.

B \times 16^2 = 11 \times 16^2 = 2816.

C \times 16^1 = 12 \times 16^1 = 192.

D \times 16^0 = 13 \times 16^0 = 13.

And add up the results:

409 + 2816 + 192 + 13 = 3430Therefore, the hexadecimal number **ABCD** is equal to the decimal number **3430**.

**For clarity watch the detailed explanatory video below**

In conclusion, converting numbers from other bases to base 10 is a useful skill for working with different number systems in mathematics and computer science. The process involves writing the number in expanded form, evaluating each term using the exponentiation rule for the base, and adding up the results. By mastering this process, we can easily convert numbers between different bases and work with them in a variety of applications.

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