Why Maths Suddenly Starts Using Letters
Learn what literals actually do, how to pick out terms, factors and coefficients, and how to tell like terms from unlike ones before you start simplifying.
What does a letter stand for in algebra?
In algebra a letter is called a literal or variable, and it stands for a number whose value can change. That is the whole reason letters appear: one statement written with a literal can describe every case at once, instead of you writing out hundreds of separate sums.
This page covers everything in the ICSE Class 6 Mathematics chapter on the fundamental concepts of algebra: what literals do, the parts of an expression, how expressions are classified, and how to write repeated multiplication in index form.
This page covers everything in the ICSE Class 6 Mathematics chapter on the fundamental concepts of algebra: what literals do, the parts of an expression, how expressions are classified, and how to write repeated multiplication in index form.
How do literals help you generalise a rule?
A literal turns a particular fact into a general one.
You know that , and that . Rather than listing every pair for ever, write it once:
That single line covers every possible pair of numbers. The same applies to formulas. The perimeter of a square with side 4 cm is , and with side 9 cm it is — so for any side ,
Literals also capture number patterns. In the sequence 3, 5, 7, 9, each term is two more than the last, and the th term can be written as .
For example, if a pencil costs Rs 5, then pencils cost rupees — one expression that prices any quantity you like.
Notice that means . In algebra the multiplication sign between a number and a literal is left out, which trips students up at first.
You know that , and that . Rather than listing every pair for ever, write it once:
That single line covers every possible pair of numbers. The same applies to formulas. The perimeter of a square with side 4 cm is , and with side 9 cm it is — so for any side ,
Literals also capture number patterns. In the sequence 3, 5, 7, 9, each term is two more than the last, and the th term can be written as .
For example, if a pencil costs Rs 5, then pencils cost rupees — one expression that prices any quantity you like.
Notice that means . In algebra the multiplication sign between a number and a literal is left out, which trips students up at first.
What are the terms, factors and coefficients of an expression?
An algebraic expression is a combination of literals and numbers joined by operations. Its parts each have a name.
Take .
Terms are the parts separated by or signs: here , and — three terms.
Variables are the literals, and . A constant is a term with no literal, so is the constant.
Factors of a term are what multiply together to make it. The factors of are 7 and .
The numerical coefficient of a term is the number multiplying the literal. In it is 7, and in it is 3.
Two cases catch people out. In the term , the coefficient is 1, not zero — means . And in the coefficient is , sign included, because the sign belongs to the term that follows it.
Take .
Terms are the parts separated by or signs: here , and — three terms.
Variables are the literals, and . A constant is a term with no literal, so is the constant.
Factors of a term are what multiply together to make it. The factors of are 7 and .
The numerical coefficient of a term is the number multiplying the literal. In it is 7, and in it is 3.
Two cases catch people out. In the term , the coefficient is 1, not zero — means . And in the coefficient is , sign included, because the sign belongs to the term that follows it.
How are expressions classified, and what are like terms?
Expressions are named by how many terms they contain. A monomial has one term (), a binomial has two (), a trinomial has three (), and polynomial is the general word for an expression with one or more terms.
Like terms have exactly the same literals raised to exactly the same powers; only their coefficients may differ. Unlike terms do not.
In , the like terms are and , and separately and .
This matters because only like terms can be added or subtracted:
But cannot be combined at all — it stays as it is.
Be careful with powers: and are unlike terms, even though both use , because the powers differ. In real life it is like adding lengths to areas — the quantities are not the same kind, so they will not combine.
Like terms have exactly the same literals raised to exactly the same powers; only their coefficients may differ. Unlike terms do not.
In , the like terms are and , and separately and .
This matters because only like terms can be added or subtracted:
But cannot be combined at all — it stays as it is.
Be careful with powers: and are unlike terms, even though both use , because the powers differ. In real life it is like adding lengths to areas — the quantities are not the same kind, so they will not combine.
Formula
How do you write repeated multiplication in index form?
When a literal multiplies itself repeatedly, write it as a power, using an index (or exponent) to count the factors:
Here is the base and the small raised number is the index. We read as "a squared", as "a cubed" and as "a to the power four".
The index counts how many times the base appears as a factor, so
Worked example: if , then .
The classic error is to multiply the base by the index instead. is not . With these differ sharply: while . Index means repeated multiplication; a coefficient means repeated addition.
Here is the base and the small raised number is the index. We read as "a squared", as "a cubed" and as "a to the power four".
The index counts how many times the base appears as a factor, so
Worked example: if , then .
The classic error is to multiply the base by the index instead. is not . With these differ sharply: while . Index means repeated multiplication; a coefficient means repeated addition.
Exam tip
The mistake most students make with coefficients and powers
Two confusions account for most lost marks here, and both come from reading a term too quickly.
First, and are completely different. means , while means . Substitute : the first is 10, the second is 25.
Second, when collecting like terms, students drop the sign in front of a term. In , treat each sign as part of its term: , , , giving .
Before simplifying anything, underline each term with its sign. It takes a moment and removes both errors at once.
First, and are completely different. means , while means . Substitute : the first is 10, the second is 25.
Second, when collecting like terms, students drop the sign in front of a term. In , treat each sign as part of its term: , , , giving .
Before simplifying anything, underline each term with its sign. It takes a moment and removes both errors at once.
Did you know
Why is the multiplication sign left out in algebra?
The symbol looks almost identical to the letter , which is the most commonly used literal of all. Writing by hand quickly becomes unreadable.
So the convention is simply to omit it: means , and means . It is a habit born of handwriting, and it is why a number written directly against a letter always means multiply.
So the convention is simply to omit it: means , and means . It is a habit born of handwriting, and it is why a number written directly against a letter always means multiply.
Key takeaways
Algebra basics in 30 seconds
- A literal is a letter standing for a number that can vary, letting one statement generalise a whole family of facts, patterns and formulas.
- Terms are separated by + and − signs; a constant has no literal, factors multiply to make a term, and the numerical coefficient is the number in front (1 when unwritten).
- Expressions are monomials, binomials, trinomials or polynomials by the number of terms.
- Like terms share the same literals to the same powers and can be combined; and are unlike, so they cannot.
- Index form counts repeated multiplication: means , which is not the same as .
You will remember all of this far better after answering five questions on it than after reading it twice.
- Terms are separated by + and − signs; a constant has no literal, factors multiply to make a term, and the numerical coefficient is the number in front (1 when unwritten).
- Expressions are monomials, binomials, trinomials or polynomials by the number of terms.
- Like terms share the same literals to the same powers and can be combined; and are unlike, so they cannot.
- Index form counts repeated multiplication: means , which is not the same as .
You will remember all of this far better after answering five questions on it than after reading it twice.