How to Change Number Bases Part 1

I have already discussed clock arithmetic,  modulo division, and number bases. We further our discussion in this post by learning how to change numbers from one base to another.

The number system that we are using everyday is called the decimal number system or the base 10 number system (deci means 10). It is believed that this system was developed because we have 10 fingers.

In the base 10 system, the digits are composed of  0 up to 9. Adding 1 to 9, the largest digit in this system, will give us 10. That is, we replace 9 in the ones place with 0, and add 1 to the tens place which is the next larger place value.

Another way to write a number in base 10 is by multiplying its digits by powers of 10 and adding them. For example, the number 2578 can be rewritten in expanded form as

2(10^3) + 5(10^2) + 7(10^1) + 8(10^0)

Suppose, we are aliens with only 5 fingers, then our digits would only compose of 0, 1, 2, 3, and 4. In other words, we will have base 5 number system. If we add 1 to 4, the largest digit in this system, then we will replace the ones digit with 0 and the next larger place value with 1 to give us 10. Note that the next larger place value in the system is the 5^1‘s or fives place and not the tens place.

In the base 5 number system, we can also write any number by multiplying its digits by powers of 5 and adding them. For example, the number 2031 can be written in expanded form

2(5^3) + 0(5^2) + 3(5^1) + 1(5^0)

Converting Base 5 to Base 10 and Vice Versa

If we are using different number bases, it is important to distinguish them from one another. Therefore, in the example above, 2031 in base 5 can be written as 2031_5. We can also write 2578_{10} to mean 2578 base 10.

If we expand any number from a different number system by multiplying its digits by its base and adding them, we will have its equivalent in base 10. For instance, if we evaluate 2031_5, we have

2(5^3) + 0(5^2) + 3(5^1) + 1(5^0) = 266_{10}.

What about converting base 10 to base 5?

We will stick with the example above, so we can check if our answer is correct. We convert 266_10 back to base 5.

The digits of numbers in base 5 are multiplied by powers of 5. That means that first, we have to check the largest power of 5 that can be subtracted from 266. We know that we cannot subtract 5^4 = 625 from 266, so the largest power of 5 that can be subtracted from 266 is 5^3 = 125. How many 125’s are there in 266? We have 2. So, we write 2 in the 5^3‘s place as shown in the table below.

change number bases

Next, we subtract 266-2(5^3) which gives us 16. We will check the largest power of 5 that can be subtracted from 15. We know that we cannot subtract 5^2 from 15, so we place 0 in the 5^2‘s place.

change number bases 2

Next, how many 5^1’s are there in 16. We have 3 since 3(5) = 15. So, we write 3 in the 5^1’s place giving us the next table.

number bases 3

Next, we subtract 16-15 = 1. Now, how many 1’s (remember 5^0 = 1) are there in 1? There’s 1. So, we write one in the 5^0‘s place or ones place.

change number bases 4

So, 266_{10} is equal to 2031 in base 5. In the next post, we will have one more example on converting number from one base to another.

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