**New** **Mathematics 1**

*UNIT 1: Sets*

A set is a group of well-defined objects.

An element of a set is one of the well-defined objects in the set.

The empty set is the set that does not contain any elements.

A Venn diagram is a closed curve which has the elements of a set inside it.

We use set parentheses to list the elements of a set.

Equal sets are sets that contain the same elements, which may be listed or described.

Equivalent sets are sets that have the same number of elements, so that a one-to-one correspondence can be set up between them.

If B is a subset of A, then all of the elements of B are in A.

A proper subset of a set is any subset of the set except the set itself.

We call the set itself an improper subset.

Equivalent sets have the same number of subsets and the same number of proper subsets.

The universal set is the set that contains all of the elements that we are talking about.

Finite sets come to an end, so their elements can be counted.

Infinite sets go on without end, so their elements cannot be counted.

The union of two sets is the set of elements that are in either of the two sets.

We call the elements in both sets common elements.

We call sets which have no common elements disjoint sets.

The intersection of two sets is the set of elements that are common to both sets.

The intersection of two disjoint sets is the empty set.

The difference between two sets is the set of elements from the first set which are not in the second set.

The complement of a set A is the difference between the universal set and set A.

*UNIT 2: Non-Metric Geometry*

Through any two different points in space there is one, and only one, line.

If two different points are in a plane than the line through them is in a plane.

An infinite number of planes pass through two points in space.

Any three points which are not in the same line are in only one plane.

A half-line is a line which goes on infinitely in one direction from an end-point which is not part of the half-line.

A ray is the union of a half-line with its end-point.

A line segment is a part of a line having an end-point at both ends each of which is a part of the line segment.

1. Through any two different points in space there is one, and only one, line.

2. If two different points are in a plane than the line through them is in a plane.

3. Any three points which are not in the same line are in only one plane.

If a point is in a line, then, the intersection of the point and the line is the point itself.

If a point is not in a line, then, the intersection of the point and the line is the empty set.

Only one plane can be drawn through a line and a point outside the line.

If a point is in a plane then the intersection of the point with the plane is the point it self.

If a point is not in a plane then the intersection of the point with the plane is the empty set.

If a line is in a plane the intersection of the line with the plane is the line itself.

If a line and a plane intersect and the line is not in the plane, then the intersection is only one point.

If the intersection of a line with a plane is empty then the line is parallel to the plane.

If two lines in a plane intersect they intersect in only one point.

If the intersection of two lines in a plane is empty then the two lines are parallel.

If two lines do not intersect and are not parallel, then, they are skew.

If two planes intersect, the intersection is a line.

If the intersection of two planes is empty then the two planes are parallel.

*UNIT 3: Natural Numbers*

Counting is to set up a one-to-one correspondence between the elements of the set that we are counting and the set of counting numbers, C= {1, 2, 3...}.

The set of natural numbers N is the set we use to show the number of elements in a set.

The set of counting numbers C is the set we use for counting the number of elements in a set.

To find how many natural numbers lie between two given natural numbers, we subtract 1 from the difference of two numbers.

A natural number raised to the power 1 is itself, a natural number (but not 0) raised to the power 0 is 1.

Addition is the operation of finding the number of elements in the union of two disjoint sets.

The set of natural number is closed under addition.

The set of natural numbers is commutative under addition.

The set of natural numbers is associative under addition.

We call the number (0) the identity element for addition of natural numbers.

The set of natural number is closed under multiplication.

The set of natural numbers is commutative under multiplication.

The set of natural numbers is associative under multiplication.

The operation of multiplication is distributive over the operation of addition in the set of natural numbers.

The identity element in the set of natural umbers under multiplication is 1.

The null element in the set of natural numbers under multiplication is 0.

The inverse operation of addition is subtraction.

Subtraction is the operation of finding a difference.

(minuend) - (subtrahend) = (difference).

The set of natural number is not closed under subtraction.

The operation of subtraction of natural numbers is non-commutative.

The operation of subtraction of natural numbers is non-associative.

Multiplication is distributive over subtraction in the set of natural numbers.

The inverse operation of multiplication is division.

Division is the operation of finding a quotient.

The set of natural number is not closed under the operation of division.

The operation of division of natural numbers is non-commutative.

The operation of division of natural numbers is non-associative.

The division of any natural number by 1 gives the natural number as the result.

We cannot divide a counting number by zero.

We cannot divide a natural number by 0.

The division of a non-zero natural number by itself gives 1 as the result.

The division of 0 by a non-zero natural number gives 0 as the result.

*UNIT 4: Prime Numbers and Factorization*

A natural number is divisible by 2 if its unit digit is an element of the set {0, 2, 4, 6, 8}.

A natural number is even if its unit digit is even.

A natural number is odd if its unit digit is odd.

A natural number is divisible by 5 if its unit digit is an element of the set {0, 5}; i.e., a natural number is divisible by 5 if its unit digit is divisible by 5.

A natural number is divisible by 3 if the sum of its digits is divisible by 3.

A natural number is divisible by 9 if the sum of its digits is divisible by 9.

A composite number is a natural number that has more than two factors.

A
prime number is divisible only by itself and 1. (see __glossary__)

Factorization: A number is factorized when it is written as the product of prime numbers.

The unique factorization property of the natural numbers: We can factories any natural number greater than one into prime factors in only one way, except for the order in which we write the factors.

*The
highest common factor* (H.C.F) of
two (or more) natural numbers is the biggest natural number which is a factor
of both (or all) of them.

Two or more natural numbers are prime to each other if their only common factor is 1.

A multiple of a given natural number is any natural number that is divisible by the given natural number.

The
product of any two natural numbers equals the product of their H.C.F with their
L.C.M. (*lowest common multiple*).

If two numbers are prime to each other than their H.C.F is 1 and their L.C.M is the product of two numbers.

*UNIT 5: Fractions*

A fraction is a part of a whole.

The denominator of a fraction is the number of equal parts that there are in the whole.

The numerator of a fraction is the number of equal parts that there are in the fraction.

If the numerator of a fraction is 1 we call the fraction a unit fraction.

A simple or proper fraction is a fraction whose numerator is smaller than its denominator, so it is less than a whole.

We call the sum of a counting number and a simple fraction a mixed number.

Improper fractions are fractions which are equal to or greater than a whole.

To change an improper fraction to a mixed number we divide the numerator by the denominator. In the division the quotient is the whole number, the remainder is the numerator and the divisor is the denominator, of the mixed number. If the denominator divides into the numerator exactly, than the improper fraction equals a counting number.

To change a mixed number to an improper fraction, we multiply the counting number by the denominator and add the numerator. This gives us the numerator of the improper fraction. The denominator of the improper fraction is the same as the denominator of the fraction part of the mixed number.

To enlarge a fraction is to multiply the numerator and denominator of the fraction by the same counting number, to get an equivalent fraction.

To reduce a fraction is to divide the numerator and denominator of a fraction by a counting number which is a factor of both. This gives a fraction which is equivalent to the first fraction.

A fraction is in its simplest from if the H.C.F. of its numerator and denominator is 1.

A *rational number* is any number that we can
write as a fraction. That is, a number that we can write as a/b where a is an element of N and b is an element of C (so b is not
equal to 0).

The bigger of two unit fractions is the one with the smaller denominator.

If two fractions have the same numerator then the biggest fraction is the one with the smaller denominator.

The sum of two rational numbers which have the same denominator is the sum of the numerators with the same denominator.

The set of rational numbers is closed under addition.

The set of rational numbers is commutative under addition.

The set of rational numbers is associative under addition.

The additive identity element in the set of rational numbers is 0.

To find the product of a natural number with a rational number, we multiply the numerator of the rational number by the natural number. This gives us the numerator of the product. The denominator of the product is the denominator of the rational number.

To multiply two fractions, we multiply the numerators together to find the numerator of the product. We, then, multiply the denominators together to find the denominator of the product.

The set of rational numbers is closed under multiplication.

The set of rational numbers is commutative under multiplication.

The set of rational numbers is associative under multiplication.

The multiplicative identity element in the set of rational numbers is 1.

The null element in the set of rational numbers under multiplication is 0.

The operation of multiplication is distributive (left to right) over the operation of addition in the set of rational numbers.

The operation of multiplication is distributive (right to left) over the operation of addition in the set of rational numbers.

(minuend) - (subtrahend) = (difference)

The difference between two rational numbers with the same denominator is the difference of the numerators with the same denominator.

To find the difference between two mixed numbers, we subtract the natural number parts and we subtract the fraction parts. We then add the two parts together.

To find the difference between two rational numbers in fraction form, first we write the two rational numbers as equivalent fractions with the same denominator. Then, we subtract the numerators and leave the same denominator.

The operation of subtraction in the set of rational numbers is neither commutative nor associative.

The operation of multiplication is distributive over the operation of subtraction in the set of rational numbers.

To find the quotient of two rational numbers, we find the product of the dividend (or numerator) with the reciprocal of the divisor (or denominator).

The quotient, when 0 is divided by a non-zero rational number, is 0.

We cannot divide by zero in the set of rational numbers.

If we divide any rational number by 1, the result is the same rational number.

If we divide 1 by any non-zero rational number, the result is the reciprocal of the rational number.

To find a fraction of a whole, we multiply the whole by the fraction.

To find the fraction that one natural number is to another non-zero natural number, we divide the first number by the second.

To find the number that a given number is a given fraction of, we divide the given number by the given fraction.

*UNIT 6: Rational Numbers in Decimal
Form*

A decimal fraction is a fraction whose denominator is a power of ten.

A terminating decimal is a decimal which has the digit zero repeating without end.

A non-terminating decimal is a decimal which has a digit different from zero, or a group of digits, repeating without end.

All rational numbers in decimal form have repeating digits.

If the denominator of a rational number in fraction form is a power of ten or can be enlarged to a power of ten, then, the repeating digit is 0 and the decimal is terminating. If the denominator cannot be enlarged to a power of ten, then, the repeating digit is different from 0, or there is a group of repeating digits, and the decimal is non-terminating.

To approximate a decimal, add 1 to the value of the last digit kept, if the first digit dropped is 5 or more, otherwise leave the digits unchanged.

The set of decimal numbers is closed under addition.

The operation of addition on the set of decimal numbers is commutative.

The operation of addition on the set of decimal numbers is associative.

The identity element for the set of decimal numbers under addition is 0.

The number of decimal places in the product equals the sum of the numbers of decimal places in the two numbers.

To multiply a decimal number by a power of 10, move the decimal point the same number of places to the right, with respect to the digits of the number, as the power of 10.

The set of decimal numbers is closed under multiplication. The operation of multiplication on the set of decimal numbers is commutative.

The operation of multiplication on the set of decimal numbers is associative.

The identity element in the set of decimal numbers under multiplication is 1.

The null element in the set of decimal numbers under multiplication is 0.

The operation of multiplication is distributive over the operation of addition in the set of decimal numbers.

The operation of multiplication is distributive over the operation of subtraction in the set of decimal numbers.

To divide a decimal number by a power of 10, move the decimal point the same number of places to the left, with respect to the digits of the number, as the power of 10.

If the prime factors of the denominator of a fraction are 2's and/or 5's, only then, the fraction gives a terminating decimal.

If the prime factors of the denominator of a fraction contain a number different from 2 or 5, then, the fraction gives a non-terminating decimal.

*UNIT 7: Angles and Triangles*

An angle is the union of two rays (sides) from the same end-point (vertex).

When two lines intersect, four angles are made.

The intersection of an angle with its interior region is the empty set. The intersection of an angle with its exterior region is the empty set. The intersection of the interior and exterior regions of an angle is the empty set.

A degree is the size of an angle that we get when we divide a complete turn into 360 equal angles; or 1 degree is 1/360 of a complete turn. A minute of angle measure is 1/60 of a degree. A second of angle measure is 1/60 of a minute.

A *complete angle* measures 360^{o}; A *straight angle*
measures 180^{o}; A *right angle*
measures 90^{o}; An *acute angle*
measures 90^{o}; An *obtuse angle*
measures between 90^{o} and 180^{o}; A *reflex angle* measures between 180^{o}
and 360^{o}.

*Congruent
angles* are angles whose degree
measures are equal.

Two angles in the same plane are adjacent (next to each other) if they have a common vertex, a common ray and no parts of their interiors intersect.

Two
angles are supplementary if the sum of their measures is 180^{o} (i.e.,
a straight angle).

Two
angles are complementary if the sum of their measures is 90^{o} (i.e.,
a right angle).

A triangle is the union of three line segments joined end-to-end in a plane.

The intersection of a triangle with its interior region is the empty set. The intersection of a triangle with its exterior region is the empty set. The intersection of the interior and exterior regions of a triangle is the empty set.

An *equilateral triangle* is a triangle in
which all three sides are of equal length.

An *isosceles triangle* is a triangle in which
two sides are of equal length.

A *scalene triangle* is a triangle in which
all three sides are of different lengths.

An *acute - angled triangle* is a triangle in
which all three angles are smaller than a right - angle (or 90^{o}).

An *obtuse - angled triangle* is a triangle in
which one of its angles is greater than a right - angle (or 90^{o}).

A *right - angled triangle* is a triangle in
which one of its angles is a right - angle (or 90^{o}).

*UNIT 8: Measurements*

Discrete sets are sets which have elements that we can count (example: number of students in a classroom, number of trees in a park).

Continuous sets are sets which have elements that we can measure, but we can't count (example: body weight, body temperature).

The more precise of two measurements is the one with the smaller unit of measure (example: 1.54 m is more precise than 1.5 m).

The greatest possible error in a measurement is a half of the unit of measure being used.

The perimeter of a figure is the sum of the lengths of the sides of the figure.

A quadrilateral is the union of four line segments joined end-to-end to form a closed figure in a plane.

A square is a quadrilateral with all of its sides equal and all of its angles right angles.

The perimeter of a square is four times the length of its side.

A rectangle is a quadrilateral with all of its angles right-angles.

The perimeter of a rectangle is twice the sum of its length and width.

One
square meter (1 m^{2}) is the area of a square whose sides are one
meter long.

One 'are' (a) is equal to one square decameter or 100 square meters.

The
units of land area: centiare (ca) = 1 m^{2}; deciare (da) = 10 m^{2};
are (a) = 100 m^{2}; dekare (daa) = 1000 m^{2}; hectare = 10.000 m^{2}.
For a list of prefixes used to indicate decimal multiplies and sub-multiples of
SI units, follow the __link__.

To measure areas we use squares of unit length.

The area (A) of a rectangle l units long and w of the same units wide is l x w square units.

If A = l x w, then l = A / w and w = A / l

The
area of a square (A) is the square of the length of its side (a): A = a^{2}.

The area of a right-angled triangle is a half of the product of its right sides: A = 1/2 x b x h.

One
cubic meter (1 m^{3}) is the volume of a cube whose edges are one meter
long.

A cube is a rectangular prism which has its three dimensions, length, width and height, equal.

The volume of a rectangular prism is the measure of how much space it takes up.

The volume of a rectangular prism is the product of the lengths of the three edges that meet at any vertex, length, width and height. It is also the product of its base area and height: V = l x b x h = B x h.

A liter is the capacity of a cubical container with each edge of length 10 centimeters, or 1 decimeter.

A
gram is the mass of one cubic centimeter of water, when it is measured at sea
level, and its temperature is 4^{o} C.

A
kilogram is the mass of one cubic decimeter of water, when it is measured at
sea level, and its temperature is 4^{o} C.

An hour (hr) is one twenty-fourth of the time it takes for the earth to turn once around its axis.

A day is the time it takes for the earth to turn once around its axis.

A year (yr) is the time it takes the earth to turn once around the sun.

*UNIT 9: Ratio and Proportion*

A ratio is a way of comparing quantities of the same kind by division.

A proportion is an equation showing that two ratios are equal to each other.

In the proportion a : b = c : d; a and d are called the extremes, b and c are called the means.

In any proportion, the product of the extremes equals the product of the means: a x d = b x c.

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 an inverse variation, as one variable increases so the other decreases and as one variable decreases so the other increases.

*Compiled by ***Dilara & M.Tevfik Dorak**