# Convert Hex to Decimal Without a Calculator

Need to convert hexadecimal numbers to decimal? There are plenty of applications and online utilities that will do the job for you, of course (such as Windows’ built-in calculator), but, particularly if you’re dealing with the kinds of nuts-and-bolts issues that can arise in computer programming or network and system configuration, there are times when it will be very much to your advantage to do the conversion on paper, or even in your head. And we will show you how to do just that.

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## First: Number Systems

Most of the time, of course, you count in the base-10 (or decimal) number system. That’s probably not news to you at all, but you may be used to thinking of the numbers that you use as just numbers, and not as part of a specific number system. But it is a system — one where each position in a multi-digit number is based on 10 raised to a higher power (i.e., multiplied by itself another time): ones, tens, hundreds, thousands, and so on to infinity.

## How Computers Count

In computer software and hardware, there are two important non-base-10 number systems: base-2 (the binary system, which we’ll cover in another post) and base-16, or hexadecimal. Base-2 is convenient for computers; they can do arithmetic and logical operations very quickly using binary numbers. But to human beings, binary numbers are just long, confusing strings of ones and zeroes. To make it easy to recognize and understand the numbers that computers use, they first need to be converted to a system that human beings find reasonably convenient.

## Making It East To Read

Base-10 would be easy to read, of course, but it doesn’t do a good job of showing things such as how computer memory is organized, or the logical boundaries between groups of binary numbers; in effect, the details get lost in translation. Base-16, on the other hand, translates very directly to and from binary, since 16 is simply 2 x 2 x 2 x 2, or 2 to the 4th power, so any multiple of 16 is also a multiple of 2. Base-16 may not be quite as easy for humans to read as base-10, but it’s close enough to do the job.

## Just For the Hex of It

Here’s how hexadecimal numbers work: Instead of counting from 1 to 10 in the one’s place, you count from 1 to 15. Then when you hit 16, you put a zero in the ones’ place, and 1 in the sixteens’ place. Of course, to do this so that it really is simple, you need to be able to represent each of the numbers from 1 to 15 using single digits (like 1, 2, 3, etc.), with no double-digit numbers (such as 11, 12, 13, etc.). The digits 1 though 9 represent the same numbers that they do in ordinary counting, but after that, you use letters, so 10 = A, 11 = B, 12 = C, 13 = D, 14 = E, and 15 = F. For 16, you use 10 — zero in the ones’ place and 1 in the sixteens’ place.

## What the Hex!

The sixteens’ place? that’s right — each position in a multi-digit hexadecimal number is based on 16 raised to a higher power, just like with base-10, only with 16. The first position is ones, of course, and the second position (corresponding to the tens) is sixteens. The third position is based on 16 x 16, or 256, so if you see 100 and you know that it’s supposed to be a hexadecimal number, you also know that it’s really 256 in base-10. And after that? 1000 in hexadecimal is 16 x 16 x 16, or 4,096 in decimal, and 10000 is 65,536; to figure out the rest of the bases, you just keep multiplying by 16.

## Do the Conversion

By now, you’re probably getting the basic idea behind converting from hexadecimal to decimal. In a nutshell, for each position in the hexadecimal number, you multiply the number at that position by the base for that position, then add the results of all of those multiplications. The best way to understand it is to step through it:

## Walking Through the Process

First, look at the E. There are two things you know about it: E is 14 in decimal, and it’s in the 1000s position, so you need to multiply it by 1000 hexadecimal, or 4,096 decimal (16 x 16 x 16). So, 14 x 4,096 = 57,344, and you’re on your way.

The next one’s easy. 1 is 1 in either system, and it’s in the 100s position, so it’s just 100 hexadecimal, or 256 decimal. Two in the 10s place shouldn’t be hard, either: 10 hexadecimal is 16, so 2 x 16 = 32. And A in the ones’ place is just plain A hexadecimal, or 10 decimal. Then you add them all up. Here’s the arithmetic:

```E12A =
E x 1000 hex = 14 x 4,096 = 57,344
1 x 100 hex = 1 X 256 =        256
2 x 10 hex = 2 x 16 =           32
A x 1 hex = 10 x 1 =            10
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So:

## Doing It On Your Own

That wasn’t hard, was it? All you need to do is take each digit, convert it to decimal, multiply it by the base for its position, then add all of the products. You can try this one yourself:

```52CD =
5 x 1000 hex = ? x ? =            ?
2 x 100 hex = ? x ? =             ?
C x 10 hex = ? x ? =              ?
D x 1 hex = ? x ? =               ?
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