1. A triangle has sides and angles. 2. A right triangle has one right angle ; that is, one angle of degrees. 3. An isosceles triangle has two angles that are equal and two sides that are equal. 4. An equilateral triangle has equal sides and equal angles. 5. Find in the above figures all the angles that are right angles; all that are less than right angles; all that are greater than right angles. 6. Cut from paper a triangle similar to the one shown in Fig. 5. Then cut it into parts as shown by the dotted lines. Re-arrange the 3 angles of the triangle as shown in Fig. 6. Compare the sum of the 3 angles with 2 right angles as shown in Fig. 6. Convince yourself that the three angles of this triangle are together equal to two right angles. 7. Cut other triangles and make similar comparisons, until you are convinced that the sum of the angles of any triangle is equal to two right angles. 1. If in figure 1, the angle a is a right angle, and the angle b is equal to the angle c, the angle b is an angle of how many degrees ? 2. If in figure 2, the angle d is an angle of 95° and the angle e is an angle of 40°, the angle f is an angle of how many degrees? 3. If in figure 3, the angle x is an angle of 75°, the angle w is an angle of how many degrees? 4. If in an oblong there are ab square feet, and the oblong is a feet long, it is feet wide. ab - a = 5. If in a rectangular solid there are abc cubic feet, and the solid is a feet long and b feet wide, it is feet thick. abc = ab = 6. Verify problems 4 and 5 by letting a = 3, b = 4, and C = 2. 7. There is a field that contains 1736 square rods; it is 28 rods long. How wide is the field ? 8. There is a solid that contains 4320 cubic inches; it is 24 inches long and 15 inches wide. How thick is the solid ? 9. How many square inches of surface in the solid described in problem 8? 10. How many inches in the perimeter of the largest face of the solid described in problem 8? PROPERTIES OF NUMBERS. To the Teacher: Under this head, number in the abstract is discussed with little or no distinction between numbers of things and pure number. It is dissociation and generalization without which there could be little progress in the "science of number” or in the "art of computation.” 146. Every number is fractional, integral,or mixed. 1. A fractional number is a number of the equal parts of some quantity considered as a unit ; as, a, .9, 5 sixths. 2. An integral number is a number that is not, either wholly or in part, a fractional number ; as, 15, 46, ninetyfive. 3. A mixed number is a number one part of which is integral and the other part fractional ; as, 57, 27.6, 2743. 147. An exact divisor * of a number is a number that is contained in the number an integral number of times. 5 is an exact divisor of 15. 148. Every integral number is odd or even. 1. An odd number is a number of which two is not an exact divisor ; as, 7, 23, 141. 2. An even number is a number of which two is an exact divisor ; as, 8, 24, 142. * Note.-An exact divisor of a number is sometimes called an aliquot part of the number. Properties of Numbers. 149. Every integral number is prime or composite. 1. A prime number is an integral number that has no exact integral divisors except itself and 1; as, 23, 29, 31, etc. 2. Is two a prime number? three ? nine ? 3. Name the prime numbers from 1 to 97 inclusive*. Find their sum. 4. A composite number is an integral number that has one or more integral divisors besides itself and 1; as, 6, 8, 9, 10, 12, 14, 15, etc. 5. Name the composite numbers from 4 to 100 inclusive. Find their sum. 6. Is eight a composite number? eleven? fifteen? (a) Find the sum of the results of problems No. 3 and No. 5. 150. To find whether an integral number is prime or composite. 1. Is the number 371 prime or composite? Operation. Explanation. 2)371 5)371 Beginning with 2 (the smallest prime 185+ 74+ number except the number 1) it is found by trial not to be an exact divisor of 371. 3)371 7)371 3 is not an exact divisor of 371. 123+ 53 5 is not an exact divisor of 371. 7 is an exact divisor of 371. Therefore 371 is a composite number, being composed of 53 sevens, or of 7 fiftythrees. t Observe that we use as trial divisors only prime numbers. If 2 is not an exact divisor of a number, neither 4 nor 6 can be. Do you see why? * There are 26 prime numbers less than 100. Properties of Numbers. 2. Is the number 397 prime or composite ? Operation. Explanation. 2)397 3)397 By trial it is found that neither 2, 3, 5, 198+ 132+ 7, 11, 13, 17 or 19 is an exact divisor of 397. 5)397 7)397 No composite number between 2 and 56+ 19 can be an exact divisor of 397: for 13)397 since one 2 is not an exact divisor of the 30+ number, several 2's, as 4, 6, 8, 12, etc., cannot be; since one 3 is not an exact 17)397 19) 397 divisor of the number, several 3's, as 6, 9, 20+ 12, etc., cannot be; since one 5 is not an exact divisor of the number, several 5's, as 10 and 15 cannot be; since one 7 is not an exact divisor of the number, two 7's (14) cannot be. No number greater than 19 can be an exact divisor of the number; for if a number greater than 19 were an exact divisor of the number the quotient (which also must be an exact divisor) would be less than 20. But it has already been proved that no integral number less than 20 is an exact divisor of 397. Therefore-397 is a prime number. 79+ 11)397 36+ 23+ Observe that in testing a number to determine whether it is prime or composite, we take as trial divisors, prime numbers only, beginning with the number two. Observe that as the divisors become greater, the quotients become less, and that we need make no trial by which a quotient will be produced that is less than the divisor. 3. Determine by a process similar to the foregoing, whether each of the following is prime or composite : 127, 249, 257, 371. 151, Any divisor* of a number may be regarded as a factor of the number. An exact integral divisor of a number is an integral factor of the number. * NOTE.— The word factor is often loosely used for integral factor. |