Squaring 103 in Your Head, and Why It Works
Learn the square and difference identities, use them to square numbers near 100 mentally, multiply number pairs instantly, and solve problems that give you a sum and a product.
Can you square $103$ without writing anything down?
Yes, in about two seconds, and the method is one identity.
Write as . Then
Long multiplication would have taken a dozen digits of working. The identity turned it into three numbers you can add in your head, because every piece is either a power of ten or a small square.
And this is not a trick for one number. Any number near a round figure yields to the same treatment, and the identities in this chapter are what make it work. This page covers the first part of the ICSE Class 8 Mathematics chapter on special products.
Write as . Then
Long multiplication would have taken a dozen digits of working. The identity turned it into three numbers you can add in your head, because every piece is either a power of ten or a small square.
And this is not a trick for one number. Any number near a round figure yields to the same treatment, and the identities in this chapter are what make it work. This page covers the first part of the ICSE Class 8 Mathematics chapter on special products.
Formula
What are the three basic identities?
Three results, each provable by ordinary bracket expansion:
Why the first is true. Expanding the brackets:
The two middle products are identical, which is where the comes from. An identity is not a new rule — it is the answer to a multiplication you have already learnt to do, written down once so you need not repeat it.
Why the third is the most useful. In the middle terms cancel, leaving only . A product of two numbers collapses into a difference of two squares, and that is what makes certain multiplications instant.
The mistake these identities exist to prevent. is not . Test it with , :
The missing is exactly . Whenever you are tempted to square a bracket term by term, substitute two small numbers and watch the answer fail.
Two rearrangements worth knowing on sight:
These let you find from a sum and a product without ever finding and themselves, which is the second great use of this chapter.
Why the first is true. Expanding the brackets:
The two middle products are identical, which is where the comes from. An identity is not a new rule — it is the answer to a multiplication you have already learnt to do, written down once so you need not repeat it.
Why the third is the most useful. In the middle terms cancel, leaving only . A product of two numbers collapses into a difference of two squares, and that is what makes certain multiplications instant.
The mistake these identities exist to prevent. is not . Test it with , :
The missing is exactly . Whenever you are tempted to square a bracket term by term, substitute two small numbers and watch the answer fail.
Two rearrangements worth knowing on sight:
These let you find from a sum and a product without ever finding and themselves, which is the second great use of this chapter.
How do you square numbers mentally with these identities?
Split the number into a round part plus or minus a small part, then apply the matching identity.
Worked example 1 — just above a round number.
Worked example 2 — just below.
The middle term is subtracted because the split used a minus sign, while the last term stays positive — . The pattern has only one negative sign, and putting a second one on the is a frequent slip.
Worked example 3 — a four-digit square.
Worked example 4 — products by the difference of squares. Multiply by .
Both are away from , so
One subtraction, no multiplication at all.
Worked example 5 — the same idea on small numbers.
When the difference-of-squares shortcut applies. The two numbers must be equally spaced about a convenient middle value. and straddle ; and straddle ; and straddle , giving .
Finding the middle is easy: it is the average of the two numbers. For the average is and each number is away, so the product is .
Worked example 6 — an algebraic difference of squares.
Here and , so and . Squaring the coefficient as well as the variable is essential — writing instead of loses the whole answer.
**Worked example 7 — expanding a squared binomial in .**
Worked example 1 — just above a round number.
Worked example 2 — just below.
The middle term is subtracted because the split used a minus sign, while the last term stays positive — . The pattern has only one negative sign, and putting a second one on the is a frequent slip.
Worked example 3 — a four-digit square.
Worked example 4 — products by the difference of squares. Multiply by .
Both are away from , so
One subtraction, no multiplication at all.
Worked example 5 — the same idea on small numbers.
When the difference-of-squares shortcut applies. The two numbers must be equally spaced about a convenient middle value. and straddle ; and straddle ; and straddle , giving .
Finding the middle is easy: it is the average of the two numbers. For the average is and each number is away, so the product is .
Worked example 6 — an algebraic difference of squares.
Here and , so and . Squaring the coefficient as well as the variable is essential — writing instead of loses the whole answer.
**Worked example 7 — expanding a squared binomial in .**
How do you solve problems that give a sum and a product?
Use the rearranged identities. You are almost never expected to find the two numbers themselves.
Worked example 1. If and , find .
A check. The numbers here happen to be and , since and . Then . The identity gave the answer without needing to spot them.
Worked example 2. If and , find .
The same answer, which is no accident: satisfies both sets of conditions, so both routes describe the same pair.
Worked example 3 — finding the product from both differences. If and , find .
Subtracting the two square identities:
The identity is worth remembering in its own right.
Worked example 4 — finding a difference of squares. If and , find .
One multiplication. Checking with , : .
Worked example 5 — the reciprocal form. If , find .
Squaring both sides:
The middle term is simply , because . So
Why this form appears so often. The product of and is always , so the awkward middle term of the expansion becomes a plain number, and the identity reduces to something you can solve in one line. The same happens for , where squaring gives .
The habit that makes all five of these routine. Write down which identity contains the quantities you have and the quantity you want. If you have a sum and a product and want a sum of squares, that is — and once the identity is chosen, the rest is arithmetic.
Worked example 1. If and , find .
A check. The numbers here happen to be and , since and . Then . The identity gave the answer without needing to spot them.
Worked example 2. If and , find .
The same answer, which is no accident: satisfies both sets of conditions, so both routes describe the same pair.
Worked example 3 — finding the product from both differences. If and , find .
Subtracting the two square identities:
The identity is worth remembering in its own right.
Worked example 4 — finding a difference of squares. If and , find .
One multiplication. Checking with , : .
Worked example 5 — the reciprocal form. If , find .
Squaring both sides:
The middle term is simply , because . So
Why this form appears so often. The product of and is always , so the awkward middle term of the expansion becomes a plain number, and the identity reduces to something you can solve in one line. The same happens for , where squaring gives .
The habit that makes all five of these routine. Write down which identity contains the quantities you have and the quantity you want. If you have a sum and a product and want a sum of squares, that is — and once the identity is chosen, the rest is arithmetic.
Exam tip
Exam tip: never square a bracket term by term
**.** With , : the left side is , the right is , and the missing is the . If ever unsure, substitute two small numbers.
In there is exactly one minus sign. The stays positive.
Square coefficients too: , not .
For mental squaring, split into a round part: ; .
For products, check whether the numbers are equally spaced about a middle value — their average. ; .
Learn the four rearrangements: , , , and .
For reciprocal problems, squaring gives a middle term of exactly , so gives .
And choose the identity before starting the arithmetic — matching what you have to what you want is the whole skill.
In there is exactly one minus sign. The stays positive.
Square coefficients too: , not .
For mental squaring, split into a round part: ; .
For products, check whether the numbers are equally spaced about a middle value — their average. ; .
Learn the four rearrangements: , , , and .
For reciprocal problems, squaring gives a middle term of exactly , so gives .
And choose the identity before starting the arithmetic — matching what you have to what you want is the whole skill.
Did you know
The difference of two squares in a rectangle
The identity can be seen rather than calculated, and once you have seen it the algebra never looks arbitrary again.
Take a square of side and cut from one corner a smaller square of side . What remains has area and is an L-shape.
Now cut that L-shape into two rectangles. One measures by ; the other measures by . Slide the second against the first along their matching edge, and the two pieces join into a single rectangle measuring by .
Nothing was added and nothing thrown away, so the area is unchanged:
The same rearrangement explains why is . A rectangle long and wide can be cut and reassembled into a square of side with a -by- square missing — and is the answer.
The identity for has a picture too: a square of side divides into a square , a square , and two rectangles each of area . That second rectangle is the that term-by-term squaring forgets.
Take a square of side and cut from one corner a smaller square of side . What remains has area and is an L-shape.
Now cut that L-shape into two rectangles. One measures by ; the other measures by . Slide the second against the first along their matching edge, and the two pieces join into a single rectangle measuring by .
Nothing was added and nothing thrown away, so the area is unchanged:
The same rearrangement explains why is . A rectangle long and wide can be cut and reassembled into a square of side with a -by- square missing — and is the answer.
The identity for has a picture too: a square of side divides into a square , a square , and two rectangles each of area . That second rectangle is the that term-by-term squaring forgets.
Key takeaways
Squares and differences of squares: quick revision
- The three identities: , , and .
- Each follows from ordinary expansion. The appears because and are the same; the middle terms cancel in the third.
- **.** At , the values are and , differing by .
- In only the middle term is negative — stays positive.
- Mental squaring: ; ; .
- Products of equally spaced numbers: ; ; ; . The middle value is the average.
- Algebraic forms: and . Coefficients get squared too.
- Sum and product problems: , gives .
- , gives — the same pair, and .
- , so gives .
- , matching .
- Reciprocal form: squaring gives a middle term of exactly , so leads to .
- The picture: cutting a -square from an -square and rearranging the L-shape gives a rectangle by — that is the identity, drawn.
Time yourself squaring ten numbers between and without writing anything. Once that is comfortable, the identities have stopped being formulas to recall and become something you simply use.
- Each follows from ordinary expansion. The appears because and are the same; the middle terms cancel in the third.
- **.** At , the values are and , differing by .
- In only the middle term is negative — stays positive.
- Mental squaring: ; ; .
- Products of equally spaced numbers: ; ; ; . The middle value is the average.
- Algebraic forms: and . Coefficients get squared too.
- Sum and product problems: , gives .
- , gives — the same pair, and .
- , so gives .
- , matching .
- Reciprocal form: squaring gives a middle term of exactly , so leads to .
- The picture: cutting a -square from an -square and rearranging the L-shape gives a rectangle by — that is the identity, drawn.
Time yourself squaring ten numbers between and without writing anything. Once that is comfortable, the identities have stopped being formulas to recall and become something you simply use.