A Note on Base Representation Theorem to crack CAT Quantitative Aptitude

CAT Quantitative Aptitude section has been known to throw questions on base representation and change very frequently. These are relatively easy questions once you know the procedure. This article details the procedure and state the generalization of the process.

Base/Radix

The number that determines the positional value of every digit in a number is called the base of that number. In a binary system, the base used is 2; in a decimal system as ours, the base used is 10. The base is also known as the Radix.

Base of a number system is used to express numbers using digits. For example, in decimal system that we us in our day to day life, the base is 10 and the system has 10 unique digits:

0, 1, 2, 3…9; similarly in a binary system we have 2 unique digits, 0 and 1 to represent any number. Let us represent a number in base 10 first, as we are familiar with this format. Take a number, say 2865. In base 10 we write its expanded notation as:

(2865)₁₀ = 2 x 10³ + 8x10² + 6 x 10¹ + 5 x 10⁰

Change of Base

If we were to express 2865 in base 2, then we need to follow the steps shown below:

Write the number by taking digits upwards as

101100110001 or (2865)₁₀ = (101100110001)₂

Alternatively. if we were to represent (101100110001)₂ into base 10, then:

Note:

For binary decimal numbers, you could have used 2^-1, 2^-2, 2^-3 etc.

Similarly, for any base the above strategy to change the base works.

Generalizing from above, we can now state the Base Representation Theorem.

Base Representation Theorem:

We all know that every natural number can be expressed in the form

a_n10ⁿ + a_n-1 10^n-1 + ...+ a₀, where ai's are whole numbers such that 0 <a_n≤9, and 0 ≤ ai ≤ 9

                                                      for i = 0, 1, …, n – 1

Here the number 10 is called the base of numeration system.

Bases other than 10 can also be used. In fact the natural numbers 2, 3, 4, 8, 16 etc., are being extensively used as bases in practical applications.

The following theorem, which we are going to state without proof, is the foundation for the use of different bases.

Base Representation Theorem

Let ‘m’ be a natural number greater than unity.

Then, every natural number n can be represented in the form:

n = akm^k + a_k-1m^k-1 + ...+ a₀, whwre k, a₀,a₁,...a_k are whole numbers such that

k≥0, 0≤ai-1<m for i = 1, 2, ....k, and 0 <ak<m