الرياضيات
What Is a Square Number?
1, 4, 9, 16, 25 and on up to 225. The name comes from a picture of dots, and the gaps between consecutive squares are the odd numbers in order.
Square number
A square number is what you get by multiplying an integer by itself, giving 1, 4, 9, 16, 25 and so on.
- يُسمّى أيضاً
- perfect square
- أين يقابله الطلاب
- Grade 4 to 6; in England, recognising square numbers and the notation for squared is a Year 5 objective in the national curriculum.
الإجابة باختصار
A square number is the result of multiplying a whole number by itself: 1, 4, 9, 16, 25, 36 and onwards. They are called square numbers because that many dots can be arranged into a perfect square — 16 dots make a block four by four. Another name for them is perfect squares.
مثال
16 dots arranged as 4 rows of 4
Try the same thing with 15 dots and it cannot be done: some row will be short. That failure is the definition made visible. A number is square exactly when its dots form a complete block with equal sides, and 16 manages it because 16 is 4 × 4.
The first fifteen, worth knowing by heart
This list does more later work than almost anything else memorised at this age. Simplifying surds, factorising quadratics, checking a Pythagoras answer and spotting scale factors all begin with recognising a number on it.
Fifteen is a sensible stopping point. Above it the squares are rarely wanted on sight, and the two that do turn up are easy enough to build: 20² is 400 and 25² is 625.
- 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25
- 6² = 36, 7² = 49, 8² = 64, 9² = 81, 10² = 100
- 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
The picture the name comes from
Square number is not jargon; it is a description of a shape. Nine counters go into a three-by-three block, sixteen into a four-by-four, twenty-five into a five-by-five. Fifteen counters cannot be arranged that way at all, which is what makes 15 not a square number.
The same picture explains the notation and the vocabulary that follow it. The area of a square is side × side, so 4² is read as four squared, and area is measured in square units for the same reason. A student who has seen the dots does not have to remember why any of that is the case.
The gaps between them are the odd numbers
Write out 0, 1, 4, 9, 16, 25, 36 and take the differences: 1, 3, 5, 7, 9, 11. The odd numbers in order, every time. So each square can be reached from the one before by adding the next odd number, and 49 + 15 = 64 is a genuine shortcut rather than a coincidence.
The dots explain it. To turn a four-by-four block into a five-by-five, you add a strip along one side, a strip along the other and one counter in the corner: 4 + 4 + 1, which is 9. That L-shaped addition is always odd, because it is two equal strips plus a single corner.
One more property is worth carrying into an exam: square numbers only ever end in 0, 1, 4, 5, 6 or 9. Any number ending in 2, 3, 7 or 8 can be ruled out at a glance. The test rules numbers out rather than in — 32 ends in 2 and is certainly not square, while 35 ends in 5 and still is not.
أسئلة شائعة
Is 0 a square number?
By the definition, yes: 0 × 0 is 0, so 0 is a perfect square. School lists usually begin at 1 because they are about counting arrangements of dots, and zero dots make an uninteresting picture. If a question asks for square numbers between two values, follow the convention your course uses.
Is 1 a square number?
Yes. 1 × 1 = 1, and a single counter is a one-by-one square. It is the first entry on the list, and it is also the only square number that is smaller than the number it came from, since squaring makes everything above 1 larger and everything below 1 smaller.
Can I tell whether a number is square just by looking?
Only sometimes, and only in the negative. A square number always ends in 0, 1, 4, 5, 6 or 9, so anything ending in 2, 3, 7 or 8 is definitely not square. Ending in one of the six digits proves nothing, since 35 and 46 both pass that test and neither is square.
What is the difference between a square number and a square root?
They are opposite directions along the same relationship. Squaring 7 gives the square number 49; taking the square root of 49 gives back 7. Learning the list of squares therefore gives you the common square roots for nothing, which is why the list repays memorising early.
المصادر
- National curriculum in England: mathematics programmes of study — Department for Education
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