New Mathematics 02
UNIT 1: Integers
The absolute value of a number is the unit distance of the graph of the number from the origin on the number line.
All positive integers are greater than 0.
All negative integers are smaller than 0.
Any positive integer is greater than any negative integer.
The bigger of two positive integers is the one with the greater absolute value.
The bigger of two negative integers is the one with the smaller absolute value.
The some of two integer shaving the same sign is the sum of the absolute values of two integers, made positive if the two integers are positive, and made negative if the two integers are negative.
The sum of any integers having different signs is the sign of the integer having the bigger absolute value in front of the difference between the absolute values of the two integers.
The sum of two integers having the same absolute value but opposite signs is zero, 0.
The integers are additive inverses of each other if they have the same absolute value and opposite signs.
The product of positive integer with another positive integer is a positive integer.
The product of a positive integer and a negative integer is a negative integer.
The product of two negative integers is a positive integer.
The product of two integers having the same sign is a positive integer.
The product of two integers having different signs is a negative integer.
If p and q are any positive natural numbers then : -
(+p ) x ( +q ) = ( +pq)
( +p ) x ( -q ) = ( -pq )
( -p ) x ( +q ) = ( -pq )
( -p ) x ( -q ) = ( +pq )
The identity element in the set of integers under multiplication is (+1)
Integers do not have multiplicative inverses.
The null element in the set of integers under multiplication is 0.
The square of any non - zero integer is a positive integer.
The cube of a positive integer is a positive integer and the cube of a negative integer is a negative integer.
A positive integer raised to any power is a positive integer.
A negative integer raised to an even power is a positive integer.
A negative integer raised to an odd power is a negative integer.
If n is the base and a is the exponent then the power : -
na where n Î Z and a Î N
is positive if n is positive
is positive if n is negative and a is even
is negative if n is negative and a is odd
If a and b are integers then : -
a - b = a + (-b)
The quotient of two positive integers is a positive integer.
The quotient when a positive integer is divided by a negative integer is a negative integer.
The quotient of two negative integers is a positive integer.
The quotient when a negative integer is divided by a positive integer is a negative integer.
The quotient of two integers having the same sign is a positive integer.
The quotient of two integers having different sign is a negative integer.
(+a) ¸ (+b) = + (a ¸ b)
( -a) ¸ ( -b) = + (a ¸ b)
(+a) ¸ ( -b) = - (a ¸ b)
(-a) ¸ (+b) = - (a ¸ b)
The division of an integer by ( +1) gives the integer as the result.
The division of zero by a non- zero integer gives zero as the result.
The division of any non-zero integer by zero is undefined.
The quotient when the integer zero is divided by the integer zero is undefined.
The division of any integer by zero is undefined.
The set of integers is not closed under division.
UNIT 2: Rational Numbers
A rational number is any number that we can writ in the form a/b, where a and b are integers, but b can't be zero.
Q= { a/b ¸ a, b Î Z, b ¹ 0 }
All integers are rational numbers.
N Ì Z Ì Q
Between every two rational numbers there are an infinite number of other rational numbers.
Rationals È Irrationals = Reals
There is a one - to - one correspondence between the set of real numbers and the set of points on the number line.
The sum of two positive rational numbers is the positive sum of the absolute values of the two numbers.
The sum of two negative rational numbers is the negative sum of the absolute values of the two numbers.
The sum of two rational numbers having the same sign is the sum of the two numbers having the same sign is the sum of the absolute values of the two numbers, made positive if the two numbers are positive, and made negative if the two numbers are negative.
The sum of two natural numbers having different signs is the sign of the number having the larger absolute value in front of the difference between the absolute values of the two numbers.
The set of natural numbers, Q, is closed under addition.
a/b Î Q and c/d Î Q, Þ (a/b + c/d) Î Q
The set of rational numbers, Q, is commutative under addition.
a/b Î Q and c/d Î Q Þ a/b + c/d = c/d + a/b
The set of rational numbers, Q, is associative under addition.
a/b Î Q, c/d Î Q, e/f Î Q Þ (a/b + c/d) + e/f = a/b + (c/d + e/f) The identity element in the set of rational numbers, Q, under addition is the number zero.
a/b Î Q, 0 Þ Q, Þ a/b + 0 = 0 + a/b = a/b
the additive inverse of any non-zero rational number is that number with the opposite sign.
a/b Þ Q and (+ a/b) + (- a/b) = 0 Þ (+ a/b) and (- a/b) are additive inverses in Q.
The product of two rational numbers having the same sign is a positive rational number.
The product of two rational numbers having the different sign is a negative rational number.
The square of any non-zero rational number is a positive rational number.
The cube of a positive rational number is a positive rational number and the cube of a negative rational number is a negative rational number.
The set of rational numbers, Q, is closed under multiplication.
a/b Î Q and c/d Î Q Þ a/b * c/d Î Q
the set of rational numbers, Q, is commutative under multiplication.
a/b Î Q and c/d Î Q Þ a/b * c/d = c/d * a/b
The set of rational number, Q, is associative under multiplication.
a/b Î Q, c/d Î Q, e/f Î Q Þ (a/b * c/d) * e/f = a/b * (c/d * e/f)
The identity element in the set of rational numbers, Q, under multiplication is the number one, 1.
a/b Î Q, 1 Î Q Þ a/b * 1 = 1 * a/b = a/b
The multiplicative inverse of any non-zero rational number is the reciprocal of the number.
a/b Î Q; a, b ¹ 0 and a/b * b/a = 1 Þ a/b and b/a are multiplicative inverses in Q.
The null element in the set of rational numbers under multiplication is zero, 0.
a/b Î Q, 0 Î Q Þ a/b * 0 = 0 * a/b = 0
The product of a rational number with (-1) gives the additive inverse of the number as the result.
a/b Î Q, -1 Î Q Þ a/b * (-1) = - a/b
the operation of multiplication is distributive over the operation of addition is the set of rational numbers, Q.
a/b Î Q, c/d Î Q, e/f Î Q Þ a/b * (c/d + e/f) = (a/b * c/d) +(a/b * e/f)
To find the difference between two rational numbers we add the minuend to the additive inverse of the subtrahend.
The set of rational numbers, Q, is closed under subtraction.
If a/b Î Q, c/d Î Q and e/f Î Q then a/b * (c/d - e/f) = (a/b * c/d) - (a/b * e/f)
The set of rational numbers under subtraction is non- commutative and non-associative.
To find the quotient of two rational numbers we multiply the dividend by the multiplicative inverse of the divisor.
The quotient when a rational number is divided by (-1) is the rational number itself.
If a/b Î Q then a/b / (+1) = a/b
The quotient when a rational number is divided by (-1) is the additive inverse of the rational number.
If a/b Î Q then a/b / (-1) = - a/b
The quotient when (+1) is divided by a non - zero rational number is the reciprocal of the rational number.
If a/b Î Q - {0} then (+1) / a/b =b/a
The quotient when (-1) is divided by a non - zero rational number is the additive inverse of the reciprocal of the rational number.
If a/b Î Q - {0} then (-1) / a/b = b/a
The quotient when "0" is divided by a non - zero rational number is "0".
If a/b Î Q - {0} then 0 / a/b = 0
The quotient when non - zero rational number is divided by a "0" is undefined.
If a/b Î Q - {0} then a/b / 0 is undefined.
The quotient when "0" is divided by "0" is undefined.
0 / 0 is undefined.
If the divisor is not zero then the set of rational number is closed under division.
The set of rational numbers under division is not commutative.
The set of rational numbers under division is not associative.
(a number between 1 and 10) * (a power of 10).
UNIT 3: Equations
The solution set of a sentence is the et which has all the solutions of the sentence and which is a subset of the replacement set of the variable.
The first degree equation with one unknown is an equation that contains only one variable and the power of this variable is 1.
If a, b and c are real numbers and a and b are equal, that is, they name the same number, then if c added both a and b, the sums will be equal.
If a, b and c are real numbers and a and b are equal, that is, they name the same number, and c is not equal to zero, then if c is multiplied by both a and b the products will be equal.
For all real numbers a, b and c;
1. If a < b then a+c < b+c
2. If a > b then a+c > b+c
When multiplying or dividing both sides of an inequality by a positive number we must not change the direction of the inequality symbol.
When multiplying or dividing both sides of an inequality by a negative number we must not change the direction of the inequality symbol.
The image of a point with respect to the x - axis is the point having the same abscissa but whose ordinate is the additive inverse of the ordinate of the point.
The image of P (a, b) with respect to the x - axis is P' ( a, - b).
The image of a point with respect to the y - axis is the point having the same ordinate but whose abscissa is the additive inverse of the abscissa of the point.
The image of P (a, b) with respect to the y - axis is P' ( -a, b).
The image of a point with respect to the origin is the point having both its abscissa and ordinate additive inverses of those of the point.
The image of P (a, b) with respect to the origin is P' ( - a, - b).
To find the image of a point in the x - axis we change the sign of the y - coordinate (ordinate).
To find the image of a point in the y - axis we change the sign of the x - coordinate (abscissa).
To find the image of a point in the origin we change the sign of both coordinates.
The graph of an equation of the form x = a, where a Î R, is a straight line parallel to the y - axis at a distance of ç a÷ units from the y - axis.
If a > 0 then the graph is to right of the y - axis, and if a < 0 then the graph is to left of the y - axis.
The graph of an equation of the form y = a, where a Î R, is a straight line parallel to the x - axis at a distance of ç a÷ units from the x - axis.
If a > 0 then the graph is to above of the x - axis, and if a < 0 then the graph is to below of the x - axis.
The graph of an equation of the form y = ax, a Î R, is a straight line passing through the origin.
UNIT 4: Ratio, Proportion and Percentages
A ratio is a way of comparing two or more quantities by division. The quantities being compared must be of the same kind and in the same units.
A proportion is a sentence showing that two ratios are equal to each other.
The product of the means equals the product of the extremes.
In a proportion, if we change the places of the means than we obtain a true statement.
In a proportion, if we change the places of the extremes than we obtain a true statement.
In a proportion, if we change the places of both the means and the extremes we obtain a true statement.
In a direct variation, as one variable increases so the other increases in the same ratio, and as one variable decreases so the other decreases in the same ratio.
In a inverse variation, as one variable increases so the other decreases in the same ratio, and as one variable decreases so the other increases in the same ratio.
We call a fraction with a denominator 100 a percentage.
Profit is the difference between the cost price and the selling price of an article when the selling price is greater than the cost price.
Loss is the difference between the cost price and the selling price of an article when the cost price is greater than the selling price.
Commission is the money a salesman gets for selling goods. It is a percent of the selling price of the goods.
Discount is the amount that the price of an article is less than the original marked price. It is a percent of the original price.
Interest is the money paid by a person who borrows money, to the lender of the money.
UNIT 5: Geometry
The sum of the angles at a point is 360o.
The sum of the adjacent angles on a line is 180o.
When two lines intersect the vertically opposite angles are equal.
Complementary angles are angles whose sum is 90o or one right angle.
Supplementary angles are angles whose sum is 180o or two right angles.
Adjacent angles on a line are supplementary.
Corresponding angles are angles which are in the same position at two different points.
Exterior alternate angles are angles on different sides of the transversal but not between the other two lines at the two point.
Alternate angles or Interior alternate angles are angles on different sides of the transversal but between the other two lines at the two point.
If a transversal intersects two parallel lines then corresponding angles are equal.
If a transversal intersects two lines and the corresponding angles are equal then the two lines are parallel.
If a transversal intersects two parallel lines then the (interior) alternate angles are equal.
If a transversal intersects two lines and the interior alternate angles are equal then the lines are parallel.
If a transversal intersects two parallel lines then the exterior alternate angles are equal.
If a transversal intersects two parallel lines and the exterior alternate angles are equal then the lines are parallel.
The triangle is a plane figure formed from the union of three line segments, whose end - points are not in a straight line.
The order of size of the sides of a triangle is the same as the order of size of the angles opposite those sides.
The biggest angle of a triangle is opposite the biggest side.
The middle size angle of a triangle is opposite the middle size side.
The smallest angle of a triangle is opposite the smallest side.
The base angles of an isosceles triangle are equal.
If two angles of a triangle are equal then the sides opposite those angles are also equal. So the triangle is isosceles.
The measures of the angles of an equilateral triangle are equal.
If the measures of the three angles of a triangle are equal then the triangle is equilateral.
The hypotenuse of a right angled triangle is the longest side.
The distance of a point from a line is the perpendicular distance which is also the shortest distance.
The sum of the measures of any two sides of a triangle is greater then the measure of the third side.
An exterior angle of a triangle is equal to the sum of the two interior opposite angles.
A parallelogram is a quadrilateral with its opposite sides parallel.
A rectangle is a parallelogram with its sides intersecting in right - angles.
A rhombus is a parallelogram with all of its sides equal.
A square is a rectangle with its sides equal.
A trapezoid is a quadrilateral with one pair of opposite sides parallel.
A deltoid is a quadrilateral in which both pairs of adjacent sides are equal.
UNIT 6: Circle and Circular Region
A circle is the set of all points in a plane that are given distance (the radius) from a given point (the center) in the plane.
The interior is the set of all points at a distance less than radius from center of the circle.
The exterior is the set of all points at a distance greater than radius from center of the circle.
A secant is a line which intersects a circle at two points.
A chord is the part of a secant whose end points are on the circle.
A diameter is part of a secant which includes the center of the circle. The end points are on the circle.
The diameter of a circle is twice the radius of the circle.
A tangent to a circle is a line which intersects the circle at one point only. The point is called the point of tangency.
An arc is part of a circle having two points of the circle as end points.
A semicircle is an arc whose endpoints are also the endpoint of a diameter to the circle.
A central angle of a circle is an angle whose vertex is the center of the circle and whose rays contain two radii of the circle.
An ac has a degree measure equal to the measure of its central angle. The units of arc measure are called degree of arc.
The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc on the circle in the segment opposite the arc.
The angle on a circle subtended by a semicircle is a right angle.
Angles subtended by the same arc in the opposite segment of a circle are equal.
If a perpendicular is drawn from the center of a circle to a chord then it bisects the chord,
The line joining the center of a circle to the mid - point of a chord is perpendicular to the chord.
Equal chords are equidistant from the center of a circle.
If two chords are equidistant from the center of a circle then they are equal.
The circumference of a circle is the measure of the distance around the circle.
The area of a circle is the area of the circular closed region bounded by the circle.
UNIT 7: Mathematical Systems
An equivalence class is a set of which has all of the elements which are equivalent to the same element in a system.
A set is closed under an operation * if for every two elements of the set, a and b, a * b is also an element of that set.
If a and b are any two elements of a set, then the operation * defined on the set is commutative if:
a * b = b * a
If a, b and c are any three elements of a set, then the operation * defined on the set is associative if:
(a * b) * c = a * (b * c)
An element cannot have an inverse unless there is an identity element in the set.
A mathematical system is a set of elements together with one or more binary operations defined on the set.
UNIT 8: Statistics and Graphs
A sample is a set of data used to show the whole data.
Compiled by Dilara & M.Tevfik Dorak