Ray's New Higher Arithmetic · Arts. 373–374 · Unit 26: Evolution
110. Roots, and why the square root method works
Goal: Your student explains roots and the radical sign, tells how many figures a root will have, and finds a square root by thinking about a real square.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
- Squared paper (optional)
What's in the book
Chapter XX, Evolution: definitions (root, index, perfect and imperfect powers, square root and cube root), squares and cubes of 1 to 9, principles about the number of figures in a power and its root, two exercises, and the first explanation of square root with a 576 sq. in. square.
The lesson, step by step
Warm-up
Ask: What number times itself makes 49? What number times itself three times makes 27?
The idea
Read:
Evolution is the process of finding roots of numbers.
Evolution is the reverse of Involution, and is sometimes called the Extraction of Roots.
The Square Root of a number is one of the two equal factors which, without any other factor, produce that number; thus, 7 × 7 = 49, and √49 = 7.
Perfect and imperfect
Read:
A perfect power is a number whose root can be exactly expressed in the ordinary notation; as 32, whose fifth root is 2.
An imperfect power is a number whose root can not be exactly expressed in the ordinary notation; as 10, whose square root is 3.1622.
How many figures?
Read:
The square of any number has twice as many, or one less than twice as many, figures as the number itself has.
So mark off a number in pairs of figures from the right; each pair gives one figure of the square root.
Work the book's example
What is the length of the side of a square containing 576 sq. in.?
The side is between 20 and 30, since 20² = 400 and 30² = 900. Take away the 20 by 20 square: 176 sq. in. are left. They form two strips 20 inches long and a small corner square, all of one width. Lay them end to end: a strip about 44 inches long. 44 × 4 = 176, so the width is 4 and the side is 24 inches.
A lesson about trying
Read:
It is important for the student to note that in the processes of evolution there must be steps of trial.
SayGuessing and checking is part of the method, not a mistake.
Exercises
Your student does the two exercises after the principles:
Prove that there will be six figures in the cube of the greatest integer of two figures.
Prove that there will be twelve figures in the fourth power of the greatest integer of three figures.
Answers to the book's problems
Exercises: the greatest integer of two figures is 99, and 99³ = 970299, six figures. The greatest of three figures is 999, and 999⁴ = 996005996001, twelve figures.
Worked out for this site from the scan. If a number in your copy differs, trust the book.