1 Arithmetic 算術The numbers 0,1,2,3are called whole numbers or integers. So 23 is an integer but 1.5 is not an integer.A number greater than zero is called a positive number.A number smaller than zero is called a negative number.0 is an integer but it is neither positive nor negative.An integer that is divisible by 2 is an even integer. An integer that is not divisible by 2 is an odd integer.For any two numbers on the number line, the number on the left is less than the number on the right, for example, 23 and -4-3.The absolute value of a number is the equivalent positive value.|+2|=+2,|-3|=+3If x is an integer divisible by another integer y, then x is said to bee a multiple of y, and y is said to be a factor or divisor of x.The factors of 12 are:1,2,3,4,6,12.The multiple of 5 are:5,10,15,20If p is a whole number which has no other factors besides itself and 1,p is a prime number. 2, 3, 5, 7, 11, are prime numbers. But 1 is not considered a prime number.The least common multiple of two numbers is the smallest number which is a common multiple of both numbers.A fraction is a number which represents a ratio or division of two numbers. A fraction is written in the form . The number on the top, a, is called the numerator; the number on the bottom, b, is called the denominator.A fraction cannot have 0 as a denominator, since division by 0 is not defined.To compare the values of two fractions. Is greater or less than ? First multiply 4 by 9 and get 36; then multiply 5 by 7 and get 35, as shown here:364535-79Since 36 is greater than 35, is greater than .(It is not necessary to multiply 7 by 9 as the common denominator.)A decimal fraction is a fraction whose denominator is some power of 10 (such as 10,100,1000 ete.)The digits of a number are named as follows:2,0 7 8. 1 4 6千百十個 十百千位位位位 分分分數數數數 位位位 數數數thousands digit 千位數hundreds digit 百位數tens digit 十位數units digit 個位數tenths digit 十分位數hundredths digit 百分位數thousandths digit 千分位數Rounding off numbers can help you get quick, approximate answers. Examples:Round off 538,423.98 to the nearest hundred. T he answer is 538,400.Round off 24.76 to the nearest tenth. The answer is 24.8.The parts of a division problem are indicated in the example below.The quotient is 5 The average or arithmetic mean of 12, 3, 6 is: (12+3+6)3 = 7.If a group of numbers is arranged in order, the middle number is called the median.To find the median of 12, 3, 6, arrange them in order: 3, 6, 12. The median is 6.If the number of items in the group is even, the median is the average of the two middle numbers. For example, to find the median of 6, 3, 30, 14, arrange them in order: 3, 6, 14, 30. The median is (6+14) 2 = 10.In general, the median and the average of a collection of numbers are different.Powers, Exponents, and Roots:If b is any number and n is a whole number greater than 0, b mean the product of n factors each of which is equal to b. Thus, b=b*b*b*b, where there are n copies of b.The expression b is called the nth power of b,n is called the exponent; b is called the base.An odd integer power of a negative number is negative, and an even integer power of a negative number is positive. For example, (-2)=-8, but (-2)=16.You should know the following laws:b=1 (b0)b=bb*b=b=b (b0) (b0) (b0)If you raise a number d to the nth power and the result is b, then d is called the nth root of b .Notice the following relations:Exercise1. All the following numbers are positive EXCEPT(A) 5.3 (B) 0.07 (C) 0 (D) 2 (E) 52. The number of the factors of 24 is(A) 9 (B) 8 (C) 7 (D) 6 (E) 53. The distinct prime factors of 24 are(A) 1,2,3 (B) 2,3 (C) 2,3,6 (D) 8,34. How many of the positive integers less than 25 are integer multiples of 4?(A) 7 (B) 6 (C) 5 (D) 4 (E) 35. The least common denominator of and is(A) (B) (C) (D) 48 (E) 966. The tenthsdigit of 3,610.48 is(A) 3 (B) 6 (C) 1 (D) 4 (E) 87. The unitsdigit of 5 is(A) 20 (B) 10 (C) 5 (D) 4 (E) 08. Round 508.64 to the nearest tenth. The result is (A) 510 (B) 508 (C) 508.6 (D) 508.7 (E) 5099. 360 divided by 37, the remainder is (A) 27 (B) 333 (C) 0 (D) 9 (E) 10. The average or arithmetic mean of 2,4,6,10 is(A) 6 (B) 5 (C) 5.5 (D) 4 (E) 311. The median of 2,4,6,10 is(A) 6 (B) 5 (C) 5.5 (D) 4 (E) 312. 1010=(A) 100 (B) 10 (C) 0.1 (D) 0.01 (E) -213. If 87 million =8.7*10, then n =(A) 5 (B) 6 (C) 7 (D) 8 (E) 914. (A) (B) 5 (C) 3 (D) 15 (E) 315. Which of the following has the greatest value?(A) (B) 0.3 (C) 0.01 (D) (E) 16. 7 less than 5x. This means(A) 7-5x (B) 5x-7 (C) 75x (E) 7-5+x17. 8 is less than 2y. This means(A) 8-2y (B) 2y-8 (C) 82y (D) 2y3, then 5+23+2. If 53, then 5-43-4.2. You can multiply or divide both sides of an inequality by the same negative quantity without altering its solution set, you must change the direction of the inequality.Iif 53, then 5(-2)3, then .Exercise 1. We know that -3. If then y=(A) +3 only (B) 3 only (C) +9 only (D) 9 only (E) +3 or -32. If P = EI , and E = IR, find P in terms of E and R. 3. If 5x+22x+5, then all of the following may be a value of x EXCEPT(A) 0 (B) 1 (C) 1 (D) 2 4. A bag of chicken feed will feed 18 chickens for 54 days. For how many days will it feed 12 chickens?(A) 36 (B) 38 (C) 54 (D) 72 (E) 815. 7x-5y = 13; 2x-7y =m 26; 9x-12y =?(A) 13 (B) 26 C) 39 (D) 42 (E) 526. How many solutions does the equation a-3 = have?(A) 0 (B) 1 (C) 2 (D) 3 (E) 4Whats the Chinese for the following word?1. integer_ 2. numerator_ 3. round off_4. decimal_5. fraction_ 6. positive_ 7. multiple_ 8. power_ 9. remainder_ 10. median_ 11. denominator_ 12. odd integer_ 13. exponent_ 14. arithmetic mean_ 15. quotient_ 16. prime number_ 17. absolute value_ 18. divisor_ 19. negative_ 20.root_ How to describe “”_ “”_GeometryAn acute angle has a measure greater than 0but less than 90.A right angle has a measure of 90.An obtuse angle has a measure greater than 90but less than 180A straight angle(which looks like a straight line ) has a measure of 180.Two angles that together have a measure of 90are called complementary angles.Two angles that together have a measure of 180(that is, two angles that together form a straight line) are called supplementary angles.If two lines intersect, the four angles formed are either equal or supplementary. The angles opposite one another are called vertical angles and are equal. The angles next to one another are supplementary.If two parallel lines are intersected by a third line, called a transversal, the eight angles formed are either equal or supplementary. In the figure, line PQ is parallel to line RS.Then angles 1 and 2 are called corresponding angles and are equal.Angles 2 and 3 are called alternate interior angles and are equal.A triangle that has two equal sides is called an isosceles triangle.If all three sides of a triangle are the same length, the triangle is called an equilateral triangle.The sum of the measures of the three interior angles of a triangle is 180.If the vertex of an angle is at the center of a circle, then the angle is called a central angle and its measure is equal to the measure of the arc it intercepts.If the vertex of an angle lies on a circle, then the angle is called an inscribed angle in the circle and its measure is equal to half the measure of the arc it intercepts.If the vertex of an angle lies on a circle, then the angle is called an inscribed angle in the circle and its measure is equal to half the measure of the arc it intercepts.In circle O, angle POQ is a central angle. Angle POQ is an inscribed angle.The number of degrees of arc in a circle is 360.The circumference of a circle with radius r is equal to 2r.Length of arc =A line segment whose endpoints are on a circle is called a chord. A chord which passes through the center of the circle is a diameter. The length of a diameter is twice the length of a radius.A line that intersects a circle at one and only one point is called a tangent. If a radius is drawn to the point at which a tangent intersects a circle, the radius is perpendicular to the tangent.A right triangle is any triangle that has a right angle. The two shorter sides of a right triangle are called the legs. The longest side, which is the side opposite the right angle, is called the hypotenuse.The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the lengths of the legs.The sides of a triangle are in the ratio :1:.The sides of a triangle are in the ratio 1:.A quadrilateral is a polygon with four sides. If the opposite sides of a quadrilateral are parallel, the figure is a parallelogram.In a parallelogram:The opposite sides are equal.The opposite angles are equal.A diagonal divides the parallelogram into two congruent triangles.The diagonals bisect each other.If all the sides of a parallelogram are equal, the figure is called a rhombus.If all the angles of a parallelogram are right angles, the figure is a rectangle.If all the sides of a rectangle are equal, the figure is a square.A quadrilateral with two parallel sides and two sides which are not parallel is called a trapezoid.The sum of the lengths of the sides of a polygon is called its perimeter.The sum of all the angles of a polygon with n sides is (n-2)180. So the sum of the angles in a quadrilateral is (4-2)180=360. The sum of the angles in a pentagon is (5-2)180=540. The sum of the angles in a hexagon is (6-2)180=720.The formulas for finding the areas and volumes of common geometric figures are show n below:Area of a rectangle, A= bhArea of a square, A = sArea of a triangle, A = Area of a parallelogram, A = bhArea of a circle, A = The volume of a rectangular solid, V = LWH.The volume of a cube, V = The volume of a right circular cylinder, V=The volume of a sphere, V = Coordinate Geometry The rectangular coordinate system consists of two perpendicular number lines that intersect at the zero point on each.The point at which the two number lines intersect is known as the origin.The horizontal number line is knows as the x-axis.The vertical number line is knows as the y-axis.The numbers to the right of the origin on the x-axis and above the origin on the y-axis are positive. The numbers to the left of the origin on the x-axis and below the origin on the y-axis are negative.The coordinates of a point P( x , y ) give the location of the point in the coordinate plane ; for example , the point P( 2 , -3 ) is located in the fourth quadrant ,2 units to the right of the y-axis and 3 units below the x-axis.To find the distance between two points P( )and Q(, use the formula: PQ = To find the midpoint of a line segment PQ, with P( ) and Q(,use the formula: M (.EXERCISE1. A picture is 6 feet wide and 8 feet long. If its frame has a width of 6 inches, what is the ratio of the area of the frame to the area of the picture?2. If angle DBG equals and angle CBE equals , then angle GBE equals 3. Four equal circles of diameter one foot touch at four points as shown in the figure. What is the area of the shaded portion (in square feet)? 4. Point P(4,2) is the midpoint of line OPC, where O is at origin (0,0). The coordinates of C are 5. The three dimensions of a rectangular solid are p, 2p and 3p . What is the total surface area of the solid? Common Verbal Problems1. DiscountA discount, usually expressed in percent, indicates the part that is to be deducted from the list price. Pay attention to the usage in English and in Chinese. For example: 10% discount -打九折 ；25% discount-打七五折Example: A bicycle originally cost $100 and was discounted 10%. After 3 months it was sold after being discounted 20%. What was the single equivalent discount?2. Interest Simple interest = principal time rateExample: How much interest will $10,000 earn in 9 months at an annual rate of 6%?3. Motion Distance traveled = rate timeExample: A man driving a distance of 90 miles averages 30 miles per hour. On the return trip he averages 45 miles per hour. What is his average speed for the round trip in miles per hour?4. Work If a worker can do a job in n days, then he can do of the job in one day.Example: It takes John an hour to do a job that Tom can do in 40 minutes. One morning they worked together for 12 minutes, then John went away and Tom finished