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# How To Find the Radix of an Equation

By Kathleen Cantor, 10 Sep 2020

A radix refers to the number of digits that are used when representing numerical figures in a positional numeral system. The maximum number of unique numbers used is called the radix. The radix can include "0." For example, the decimal system contains 10 unique digits from 0 to 9. The radix of the decimal system, therefore, is 10, or Base 10.

How to find the radix of an equation is not quite the same as finding the radix of a number in a positional numbering system. Finding the radix of an equation is more like looking at an equation and determining the radix in which the calculations are done.

Normally, to find the radix of a number all you have to do is look at its subscript. The number written is its radix, or base. For example (a) b means that the number "a" is written in terms of "b." This number has "b" number of unique digits in its system.

1. Decimal Numeral System (10): If you are reading this then you must have used the decimal system in your life. This is the most popular number system in the whole world! It is used in everyday mathematics, mechanical counters, and arithmetic. It ranges from 0 to 9 and has 10 digits.
2. Binary Numeral System (2): Like you must have already guessed, this is a series of zeros and ones that are used in most computers and phones. It contains just two digits which are 0 and 1.
3. Octal Numeral System (8): The Octal system is occasionally used for computer systems because of the shorthand it provides for binary. It's eight numbers range from 0 to 7 and represent 3 bits (23).
4. Sexagesimal Numeral System: This numbering system has an interesting history that runs as far back as the Babylonians. It is still used today in concepts of minutes, seconds, and degrees.
5. Duodecimal Numeral System (12): This system is mostly used in dozens and grosses. It is easier to work with and manipulate because of the way 2, 3, 4, and 6 can easily go into them.
6. Hexadecimal Numeral System (16): Hexademical is also used as a shorthand for binary. Each of its digits works with a sequence of four binary digits. Because it's greater than the decimal, the rest are represented by "a" to "f."

There is no general or right way to find the radix of an equation. You can use various elimination methods without solving the equation to figure out what the radix might be.

For instance, an equation containing the number 9 means that all the all radices below base 10 (decimal numeral system) are eliminated.

If you see 11 or 10 in an equation, it may be part of the decimal or binary numeral system -- or any numbers system for that matter. But when you see an equation like 11 + 10 = 21, you automatically realize that because of the presence of 2 (21), binary is eliminated.

## Examples

### Example One

x2 – 3x - 10 = 0. Roots are 5 and -2

If you are familiar with quadratic equations, may know that the standard expression is ax2 + bx + c. We will be using this below.

Solution

Let’s assume that m is the first root and n is the second root so:

m + n = - b/a

m + n = b

m x n = c/a

Now, substituting with our actual quadratic equation:

m + n = - (-3)/1

m + n = 3

m = 5 and n = -2

5 x (-2) = 10/1

1010 = 10r

1 x 10 + 0 = 1 x R + 0

R = 10

Since the result is 10, is means the radix is decimal.

### Example Two

Find the radix of the equation 137 + 144 = 303.

Solution

For an equation like this, you can look at the digits and immediately eliminate binary. Now, remove the first two digits of each number.

7 + 4 = 3

In decimal, this should equal 11, so decimal is eliminated as well. But if you subtract 3 from 11, you end up with 7. This is the highest digit available in the octal numbering system. This narrows it down. To confirm, this will mean the value 137 is 95, 144 is 100, and 303 is 195 in decimal so everything fits.

### Example Three

Find the radix of the equation 106 + 74 = 202

Solution

For equations as simple as this, you can use another approach. Note: you can’t use this method on complex figures. Notice how all numbers in a particular base can be broken down in a particular way? 453610 can also be written as

X1Rn−1 + X2Rn−2 + … + XnR0,

Where R is the radix, X is the digit at a particular position and n is the number of digits contained in the value.

(4 x 103) + (5 x 102) + (3 x 101) + (6 x 100)

The general equation is:

(1a2 0a +6) + (7a + 4) = 2a2 + 0a + 7

To find the radix, solve for a and radix will equal a + 1.

A2 +7a + 10 = 2a2 + 7

10 -7 = 2a2 –a2 + 7a

A2 + 7a – 3 = 0

Solving this equation, a will be roughly = 0 or 7.  This means the radix is octal.

The more you solve for the radix of equations the more you will start developing your own way of solving for it.

## Conclusion

You may notice that most radices are natural numbers. That does not mean that other positions cannot be found. The golden ratio base and negative radix are excellent examples of this.

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