A Square and A Cube · Lesson 1 of 4
Square Numbers
“Explore square numbers, perfect-square patterns, odd-number relationships, triangular numbers and square roots.”
• Explain what a square number is and connect it with the area of a square. • Explain why perfect squares have an odd number of factors. • Use patterns in units digits, zeros and odd numbers to study perfect squares. • Connect perfect squares with consecutive odd numbers and triangular numbers. • Find and estimate square roots using known squares and prime factorisation.
Imagine 100 closed lockers numbered 1 to 100. Person 1 toggles every locker, Person 2 every second locker, Person 3 every third locker, and so on. A locker is toggled once for every factor of its number. Most factors come in pairs: for 6, the pairs are 1 × 6 and 2 × 3. So locker 6 is toggled four times and ends closed.
Square numbers behave differently. For 16, the factor pair 4 × 4 uses the same factor twice. The factors are 1, 2, 4, 8 and 16 — five factors in all. That is why square-numbered lockers are toggled an odd number of times and remain open. Among lockers 1 to 100, the open lockers are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
A number that can be written as the product of a number with itself is called a square number. The squares of natural numbers are called perfect squares.
The word square comes directly from geometry. A square with side 4 units has 4 rows and 4 columns of unit squares, so its area is 4 × 4 = 16 square units. In the same way, a square with side n units has area n × n.
Problem
Find 13².
- 1.13² means 13 × 13.
- 2.13 × 13 = 169.
- 3.Therefore, 13² = 169.
Problem
Find (3/5)².
- 1.Square both the numerator and denominator.
- 2.(3/5)² = (3 × 3)/(5 × 5).
- 3.(3/5)² = 9/25.
Problem
Find (2.5)².
- 1.Write the square as multiplication: 2.5 × 2.5.
- 2.2.5 × 2.5 = 6.25.
- 3.Therefore, (2.5)² = 6.25.
Patterns and Properties of Perfect Squares
Look at the units digits of the first several perfect squares. A perfect square can end only in 0, 1, 4, 5, 6 or 9. Therefore, any natural number ending in 2, 3, 7 or 8 cannot be a perfect square.
| Units digit of a number | Units digit of its square |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 6 |
| 5 | 5 |
| 6 | 6 |
| 7 | 9 |
| 8 | 4 |
| 9 | 1 |
This test works in only one direction. If a number ends in 2, 3, 7 or 8, it is definitely not a perfect square. But ending in 0, 1, 4, 5, 6 or 9 does not prove that a number is a square. For example, 36 is a square, but 26 is not, even though both end in 6.
Problem
Can 2038 be a perfect square?
- 1.2038 ends in 8.
- 2.No perfect square ends in 8.
- 3.Therefore, 2038 cannot be a perfect square.
Trailing zeros give another useful pattern. If a whole number ends with one zero, its square ends with two zeros. If it ends with three zeros, its square ends with six zeros. In general, squaring doubles the number of trailing zeros. So a perfect square cannot have an odd number of trailing zeros.
Parity also stays the same when a whole number is squared: the square of an even number is even, and the square of an odd number is odd.
Perfect Squares and Odd Numbers
Consecutive perfect squares grow by consecutive odd numbers. Starting with 1, add 3 to get 4, add 5 to get 9, add 7 to get 16, and so on. Visually, each larger square is made by adding an L-shaped strip of unit squares around the previous square.
Problem
Given that 35² = 1225, find 36² without multiplying 36 × 36.
- 1.The 36th odd number is 2 × 36 − 1 = 71.
- 2.The next square is obtained by adding the next odd number.
- 3.36² = 35² + 71 = 1225 + 71 = 1296.
The same pattern can be used in reverse. Repeatedly subtract 1, 3, 5, 7, … from a number. If you reach exactly 0, the number is a perfect square. If you cross below 0, it is not.
Problem
Show that 25 is a perfect square.
- 1.25 − 1 = 24.
- 2.24 − 3 = 21, then 21 − 5 = 16.
- 3.16 − 7 = 9, then 9 − 9 = 0.
- 4.We reached 0 after subtracting the first five odd numbers, so 25 = 5².
Perfect Squares and Triangular Numbers
Triangular numbers can also combine to make squares. The chapter shows the pattern 1 + 3 = 4, 3 + 6 = 9, and 6 + 10 = 16. In each case, two consecutive triangular numbers fit together to make the next square number.
| Two consecutive triangular numbers | Sum | Square |
|---|---|---|
| 1 and 3 | 1 + 3 = 4 | 2² |
| 3 and 6 | 3 + 6 = 9 | 3² |
| 6 and 10 | 6 + 10 = 16 | 4² |
| 10 and 15 | 10 + 15 = 25 | 5² |
Square Roots
A square root reverses the operation of squaring. If the area of a square is 49 cm², then its side is 7 cm because 7 × 7 = 49. We therefore say that 7 is the positive square root of 49.
If y = x², then x is a square root of y. In this chapter, we use the symbol √y for the positive square root.
Both 8² and (−8)² equal 64, so +8 and −8 are integer square roots of 64. However, in this chapter, √64 means the positive square root, 8.
For larger numbers, prime factorisation gives a reliable test. A perfect square has prime factors that can be grouped completely into pairs. Taking one factor from each pair gives the positive square root.
Problem
Find √576.
- 1.576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3.
- 2.Group equal factors in pairs: (2 × 2)(2 × 2)(2 × 2)(3 × 3).
- 3.Take one factor from each pair: 2 × 2 × 2 × 3 = 24.
- 4.Therefore, √576 = 24.
When a number is not a perfect square, we can still estimate its square root by locating it between nearby perfect squares. For example, 225 = 15² and 256 = 16², so √250 lies between 15 and 16. Since 250 is much closer to 256 than to 225, √250 is close to 16, but slightly less.
Quiz
Which number is definitely not a perfect square?
What is the next perfect square after 20² = 400?
Which statement is always true?
The prime factorisation of a number contains 2⁴ × 3² × 5². What can you conclude?
Between which two consecutive integers does √90 lie?
Practice Problems
- Without calculating square roots, identify which of 1232, 1521, 1877 and 2304 can be ruled out as perfect squares using only their units digits.
- Given 42² = 1764, find 43² using the consecutive-odd-number pattern.
- Use prime factorisation to decide whether 1156 is a perfect square. If it is, find its square root.
- Find the smallest natural number by which 675 must be multiplied to make the product a perfect square. Then find the square root of the product.
- Estimate √190 by locating 190 between two consecutive perfect squares. State the two integers between which the square root lies.
Key Takeaways
• A square number has the form n² = n × n. • Perfect squares are the only natural numbers with an odd number of factors. • A perfect square can end only in 0, 1, 4, 5, 6 or 9, but these endings do not by themselves prove that a number is a square. • Consecutive perfect squares differ by consecutive odd numbers. • The sum of the first n odd numbers is n². • In prime factorisation, every prime factor of a perfect square can be paired. • Square root reverses squaring; this chapter uses the positive square root.
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Cubic Numbers