Study Notes – Mathematical Signs
(Prepared for NMMS – Mental Ability Test, with an IIT‑JEE focus)
1. Overview
In competitive exams you will encounter a wide variety of mathematical signs (or symbols) that convey relations, operations, or properties of numbers.
A strong grasp of these signs and the algebraic rules that govern them will let you:
- Evaluate expressions quickly.
- Solve inequalities and equations efficiently.
- Spot “trick” questions that rely on subtle sign conventions.
Below we give a systematic, exam‑oriented treatment: definitions, derivations, key formulas, and pitfalls.
2. Basic Arithmetic Signs
| Symbol | Common Name | Meaning | Example |
|---|---|---|---|
+ |
Plus / Addition | Add two numbers | 5 + 3 = 8 |
- |
Minus / Subtraction | Subtract second number from first | 7 - 4 = 3 |
× or * |
Multiplication | Product of two numbers | 2 × 6 = 12 |
÷ or / |
Division | Quotient of two numbers | 12 ÷ 3 = 4 |
= |
Equality | Two expressions represent the same value | 2+3 = 5 |
≠ |
Not equal | Two expressions are different | 3 ≠ 4 |
< |
Less than | First is smaller | 2 < 5 |
> |
Greater than | First is larger | 7 > 4 |
≤ |
Less than or equal to | ≤ | 4 ≤ 4 |
≥ |
Greater than or equal to | ≥ | 6 ≥ 5 |
Tip 1 – “Minus” vs “Negative”
The minus sign-is an operation (subtraction).
The negative sign-is a property of a number (e.g.,-3).
In an expression-x, the minus sign is outside the variable; the sign ofxmay change the result.
3. Sign of Products and Quotients
3.1 Sign Table
sgn(a) |
sgn(b) |
sgn(a·b) |
sgn(a/b) |
|---|---|---|---|
| + | + | + | + |
| + | – | – | – |
| – | + | – | – |
| – | – | + | + |
| 0 | any | 0 | undefined (if b=0) |
3.2 Derivation
-
Multiplication:
[ a\cdot b = |a||b|\cdot sgn(a),sgn(b) ]
Since (|a||b| > 0) (unless one factor is zero), the sign is simply the product of the signs. -
Division:
[ \frac{a}{b} = \frac{|a|}{|b|}\cdot \frac{sgn(a)}{sgn(b)}
]
The sign is the ratio of the signs.
Trick Question – “What is the sign of ( (−3)(−2) ) ?”
Many students forget that two negatives make a positive.
4. Sign of Exponentiation
4.1 Integer Exponents
| Base | Exponent n |
Sign of a^n |
|---|---|---|
>0 |
any integer | + |