CAT 2025 Lesson : Number Systems - Concepts & Cheatsheet

1) Roman Numerals: Following are the symbols used

Symbol	I	V	X	L	C	D	M
Base $9$	$1$	$5$	$10$	$50$	$100$	$500$	$1000$

In the Roman Numeral system,
(a) A lower-valued symbol is subtracted if written to the left of its immediate higher-valued symbol.
(b) If not, the value of the number is added.

2) In a number system of base b,
(a) There are b digits.
(b) The place values to the left of the point are $b^{0}, b^{1}, b^{2},$ ...
(c) The place values to the right of the point are $b^{-1}, b^{-2}, b^{-3},$ ...

3) To convert a number from base b to base $\bm{10}$, we multiply the face values with the respective place values and add the values.

4) To convert a number from base $\bm{10}$ to base b, successively divide the number by b till we get a quotient of $0$. The remainders from bottom to top are the digits of the number (in base $10$) from left to right.

5) If $a$ and $b$ are the two bases and $a^{n} = b$, then each digit of the number in base $b$ has $n$ digits in base $a$.

6) If two or more numbers and their sum/product is given in a particular base, we compare the sum/product with that in the decimal system and deduce the base.