2.3 Converting between number systems

Let’s break it down by analyzing the value of each digit’s position. Every number has a specific “worth” based on its position.

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Understanding place value in binary

In the decimal system, the first digit (the digit in the rightmost position) represents 10⁰ = 1, the second digit represents 10¹ = 10, the third represents 10² = 100, and so on.

In the binary system, each place value is a power of 2. In binary, the value of each “place” increases by powers of 2, starting from the rightmost digit (just like in decimal where it starts with powers of 10).

Here’s how it works:

• The rightmost place in a binary number is worth 1 (2⁰ = 1).
• The second place is worth 2 (2¹ = 2).
• The third place is worth 4 (2² = 4).
• The fourth place is worth 8 (2³ = 8), and so on.

This means that each digit in a binary number has a place value that is a power of 2, and the value of the number is the sum of these powers, where the digit is either 1 (on) or 0 (off).

numerical value	binary	decimal
20	0001	1
21	0010	2
22	0100	4
23	1000	8

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Converting a Binary Number to Decimal

Let’s take a binary number and break it down to see how the place values work.

Consider the binary number 1011:

• The rightmost digit (1) is in the 1st place, and it’s worth 1. So, this represents 1 × 2⁰ = 1.
• The next digit (1) is in the 2nd place, and it’s worth 2. So, this represents 1 × 2¹ = 2.
• The next digit (0) is in the 3rd place, and it’s worth 4, but since the digit is 0, it contributes 0 × 2² = 0.
• The leftmost digit (1) is in the 4th place, and it’s worth 8. So, this represents 1 × 2³ = 8.

Now, adding up all these values gives us the decimal equivalent:

8 + 0 + 2 + 1 = 11

So, 1011 in binary is equal to 11 in decimal.

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General conversion

For any binary number, you can calculate its decimal value by summing the products of each digit and its corresponding place value (which is a power of 2). The key difference between binary and decimal is that binary works with powers of 2 instead of powers of 10.

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Consider the binary number 1101:

• 1 × 2³ = 8
• 1 × 2² = 4
• 0 × 2¹ = 0
• 1 × 2⁰ = 1

Adding them together:

8 + 4 + 0 + 1 = 13. So 1101 in binary equals 13 in decimal.

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Consider the binary number 10100:

• 1 × 2⁴ = 16
• 0 × 2³ = 0
• 1 × 2² = 4
• 0 × 2¹ = 0
• 0 × 2⁰ = 0

Adding them together:

16 + 0 + 4 + 0 + 0 = 20. So 10100 in binary equals 20 in decimal.

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Converting a Hexadecimal to Decimal

Let’s consider a hexadecimal number, the procedure is the same but the base is different. Instead of using a base of 2 (binary) or 10 (decimal), hexadecimal uses a base of 16.

To represent 16 different values in a single digit, hexadecimal uses both numbers and letters:

Hexadecimal Digits:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Here, A = 10, B = 11, C = 12, D = 13, E = 14, F = 15 in decimal.

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Consider the Hexadecimal number 1F:

• 1 × 16¹ = 1 × 16 = 16
• F × 16⁰ = 15 × 1 = 15

Adding them together:

16 + 15 = 31. So 1F in hexadecimal equals 31 in decimal.

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Consider the Hexadecimal number 010:

• 0 × 16² = 0 × 256 = 0
• 1 × 16¹ = 1 × 16 = 16
• 0 × 16⁰ = 0 × 1 = 0

Adding them together:

0 + 16 + 0 = 16. So 010 in hexadecimal equals 16 in decimal. Note that leading zeros do not affect the value.

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Consider the Hexadecimal number 38:

• 3 × 16¹ = 3 × 16 =48
• 8 × 16⁰ = 8 × 1 = 8

Adding them together:

48 + 8 = 56. So 38 in hexadecimal equals 56 in decimal.

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This place-value system is the foundation of how computers process information. Since binary uses only 0s and 1s, each digit is like a simple switch that either contributes or doesn’t contribute to the total value, depending on its position. When you stack up multiple bits, you get a larger range of values that computers use to represent everything.

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