Algebraic Division.

143. PROBLEMS.

1. Divide 4 a3x + 8 a2x2 + 6 ax3 by 2 ax.

2. Multiply the quotient of problem 1, by 2 ax.

3. Verify problems 1 and 2, by letting a = 2, and x = 3.

4. Divide 3 ab3 + 6a2b2 + 9 a3b by 3 ab.

5. Multiply the quotient of problem 4, by 3 ab.

6. Verify problems 4 and 5, by letting a = 3, and b = 5.

7. Divide 2xy + xy-xy3 by xy.

8. Multiply the quotient of problem 7, by xy.

9. Verify problems 7 and 8, by letting x = 2, and y = 3.

11. Multiply the quotient of problem 10, by a3y.

12. Verify problems 10 and 11, by letting a = 1, and y = 2.

13. Divide 3 b*x + b3x2 − 3 b2x3 by bx.

14. Multiply the quotient of problem 13, by bx.

15. Verify problems 13 and 14, by letting b= 3, and x = 4.

Observe that when the divisor is a positive number, each term of the quotient has the same sign as the term in the dividend from which it was derived.

2)86 4 - 3

One half of + 8 is + 4; one half of - 6 is – 3.

16.

2x)4x3 − 6 x2 + 8 x3 − 2 x2 + 6x.

17. Verify by letting = 2.

Geometry.

144. TRIANGLES.

a

of

2. A right triangle has one right angle; that is, one angle

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 ƒ 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 =

feet

5. If in a rectangular solid there are abc cubic feet, and the solid is a feet long and b feet wide, it is 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, 8, 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, 5, 27.6, 274§.

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.

5 is not an exact divisor of 1.5. 163 is an exact divisor of 100.

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?

123+

53

Explanation.

Beginning with 2 (the smallest prime number except the number 1) it is found by trial not to be an exact divisor of 371.

3 is not an exact divisor of 371.

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. †

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.

+ Note the similarity of the words composed and composite.