Concepts: A Square and a Cube (part 1)

1.1 From lockers to squares

The chapter opens with a puzzle. One hundred lockers are numbered 1 to 100. The first person opens each locker, the second flips lockers 2, 4, 6 and so on, the third flips multiples of three, and the pattern continues. A locker ends open exactly when it is toggled an odd number of times. The number of toggles of locker n equals the number of factors of n. Most factors come in pairs, so most lockers are toggled an even number of times and end closed. A square number has one factor that pairs with itself, so its factor count is odd. The open lockers are therefore the ten squares 1, 4, 9 and 16, then 25, 36, 49, 64, 81 and finally 100.

1.2 What a square number is

A number obtained by multiplying a natural number by itself is a square number. The square of n is written n^2 and read as "n squared". Counting squares of counting numbers produces the perfect squares; the first few are one, four, nine, sixteen and twenty-five. The word square is used because the area of a square of side n units is n x n = n^2 square units. For example, a square of side 7 cm has area 49 sq cm.

This definition also works for fractions and decimals. A square of side 3/5 units has area (3/5)^2 = 9/25 square units, and a square of side 2.5 units has area (2.5)^2 = 6.25 square units.

1.3 Patterns in the units digit

Look at the first ten squares: 1, 4, 9, 16, 25 and then 36, 49, 64, 81 and 100. Their units digits are 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. Only 0, 1, 4, 5, 6 and 9 appear. No perfect square ends in 2, 3, 7 or 8. This gives a quick test: a number ending in 2, 3, 7 or 8 is certainly not a square.

The test does not work the other way. A number ending in 6 can be a square, like 16 or 36, but 26 is not a square. So the units digit can rule a number out, not rule it in.

A square of a number ending in 1 or 9 ends in 1; a square of a number ending in 2 or 8 ends in 4; a square of a number ending in 3 or 7 ends in 9; a square of a number ending in 4 or 6 ends in 6; and a number ending in 5 has a square ending in 25.

1.4 Zeros and parity

If a number ends in one zero, its square ends in two zeros: 10^2 = 100, 20^2 = 400. If it ends in three zeros, its square ends in six zeros. In general, the square has twice as many trailing zeros as the number. So a square can never end in exactly one zero, or in exactly three zeros.

The square of an even number is even and the square of an odd number is odd. This parity rule helps eliminate options quickly.

1.5 Squares from odd numbers

The gaps between consecutive squares are consecutive odd numbers: the jump from 1 to 4 is 3, from 4 to 9 is 5, from 9 to 16 is 7, and from 16 to 25 is 9. Going the other way, adding odd numbers from 1 gives squares: 1 = 1^2, 1 + 3 = 2^2, 1 + 3 + 5 = 3^2, 1 + 3 + 5 + 7 = 4^2.

The total of the first n odd numbers always lands on n^2, and the nth odd number is 2n - 1. This gives a fast way to move from one square to the next: from 35^2 = 1225, add the 36th odd number 71 to get 36^2 = 1296.

1.6 Square roots

If y = x^2, then x is a square root of y. Each positive perfect square comes with a pair of integer roots, a positive one and its negative twin, because both 8 x 8 and (-8) x (-8) equal 64. The square root symbol is the radical sign. We write the positive root as the square root of 64 equals 8, and write both roots as plus or minus 8.

In this chapter we use the positive square root unless stated otherwise. A square root answers a natural question: what side length gives this area? A square of area 49 sq cm has side 7 cm because 7^2 = 49.

1.7 Worked examples

Example 1: Squares of fractions and decimals. Find (3/5)^2 and (-0.6)^2. Multiply each number by itself: (3/5) x (3/5) = 9/25, and (-0.6) x (-0.6) = 0.36. The fraction 9/25 is positive, as is 0.36, because a negative number squared is positive.

Example 2: Using the odd-number gap. Given 41^2 = 1681, find 42^2. The next odd number after the first 41 odd numbers is 2 x 42 - 1 = 83. So 42^2 = 1681 + 83 = 1764. Check: 42 x 42 = 1764.

Example 3: Deciding whether a number is a square. Is 1027 a perfect square? It ends in 7, and no perfect square ends in 7. So 1027 is not a perfect square. Is 1296 a square? It ends in 6, so the units digit does not rule it out. Since 36^2 = 1296, it is a square.