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What Is a Constant in Math?
A constant is a value that does not change. In the expression 5x + 3 the 3 is a constant: whatever x turns out to be, the 3 stays 3. This guide covers constants inside expressions, the named constants like pi and e, how a constant differs from a variable and from a coefficient, and what the constant term does to a graph. The simplify calculatorhandles expressions with both kinds of part.
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What is a constant in math?
A constant is a value that does not change. In 5x + 3 the 3 is a constant: whatever x turns out to be, the 3 is still 3.
That is the whole idea, and it is defined by contrast. A variable is a symbol standing for a value that can change; a constant is a value that cannot. Algebra is largely the business of working out what the variables can be while the constants stay where they are.
Constant vs variable vs coefficient
The four words that describe the parts of an algebraic expression are best learned together, because each is defined partly by the others. Take 5x + 3:
| Part | In 5x + 3 | What it is |
|---|---|---|
| Term | 5x and 3 | A piece separated by + or − |
| Coefficient | 5 | The number multiplying the variable |
| Variable | x | The letter standing for an unknown |
| Constant | 3 | A fixed number with no variable |
The constant is the part this page is about. The companion guides cover what a term is, what a coefficient is, what a variable is and what a constant is — between them they name every piece of 5x + 3 exactly once.
Named constants
Some constants come up often enough to have earned a symbol of their own, so that nobody has to write the digits out.
- π is the ratio of a circle's circumference to its diameter, about 3.14159. Every circle gives the same ratio, which is what makes it a constant rather than a measurement.
- e is about 2.71828 and is the base of the natural logarithm. It turns up wherever something grows in proportion to its own size.
- i is defined by i² = −1, which no real number satisfies.
These are irrational in the first two cases, meaning their decimal expansions never end and never repeat. That does not make them any less constant: π has exactly one value, and the fact that we cannot write it all down is a limitation of notation rather than of the number.
The constant term on a graph
In a linear equation written y = mx + b, the constant term b has a visible job: it is the y-intercept, the height at which the line crosses the vertical axis.
For y = 5x + 3, setting x = 0 gives y = 3, so the line passes through (0, 3). Change the constant to 8 and the whole line slides up five units without tilting; change the coefficient instead and it tilts without sliding. The two parts do different things and it is worth seeing that they are independent.
A constant function is the extreme case: f(x) = 4 has no variable term at all, so its graph is a horizontal line at height 4 and the output is 4 for every input.
When a letter is a constant
Confusingly, a constant does not have to look like a number. In y = mx + b, both m and b are letters, yet both are constants: they hold still while x and y vary.
The distinction is about ROLE, not about whether the symbol is a digit or a letter. Within one problem, whatever is fixed is a constant and whatever moves is a variable. Letters from the start of the alphabet are conventionally used for constants and letters from the end for variables, which is why ax² + bx + c reads unambiguously to anyone who knows the convention.
How to spot the constant
Scan the expression for a term with no letter attached. That is the constant.
- 5x + 3 → constant 3
- 5x − 7 → constant −7
- 2x² + 4x → constant 0, not written
- 9 → constant 9, the whole expression
The third case is the one people miss. An expression with no visible constant has a constant of 0, because 2x² + 4x and 2x² + 4x + 0 are the same thing. That matters when a question asks for the y-intercept: it is 0, not absent.
Common mistakes
- Dropping the sign. In 5x − 7 the constant is −7, not 7.
- Calling the coefficient the constant. Both are fixed numbers, but the coefficient multiplies a variable and the constant term does not.
- Assuming a letter must be a variable. In y = mx + b, m and b are constants.
- Treating π as 3.14 exactly. It is a constant, but 3.14 is a rounded version of it.
Quick summary
- A constant is a value that does not change.
- In 5x + 3 the constant term is 3; the 5 is a coefficient.
- π and e are named constants with one fixed value each.
- A letter can be a constant if it holds still within the problem.