Numbers in Base 2 are real numbers that can be involved in mathematical operations in the same way as more familiar decimal or Base 10 numbers. The advantage of binary systems — that they contain two numerals 0 and 1 — means that they can be used in situations that have two alternatives.

Base 2, or binary, is the smallest grouping possible. In Base 2, 0 means 0, 1 means 1, and 10_{2 }means 2, 11_{2 }means 3, 100_{2} means 4, 101_{2 }means 5, 110_{2 }means 6, 111_{2 }means 7, 1000_{2 }means 8, 1001_{2 }means 9, and 1010_{2 }means 10 in the decimal (Base 10) system. Notice that place value follows the same pattern in both bases. 10^{0} is 1, and 2^{0} is 1, 10^{1} is 10 and 2^{1} is 2, 10^{2} is 100 and 2^{2 }is 4, represented in Base 2 as 100_{2}; 10^{3} is 1000, and 2^{3 }is 8, represented in Base 2 as 1000_{2}. The difference is that a number such as 10_{2} (which means 2) is read either as “two” or “one-zero”, 100_{2} is read as “eight” or “one-zero- zero”, and so on.

Binary numbers add, subtract, multiply, and divide according to the same rules as numbers in any other base. Addition of two binary numbers involves carrying, similar to adding numbers in Base 10. Suppose the numbers to be added are 110011_{2} and 10001_{2}. Adding from right to left, the sum is 1000100_{2}, because adding 1 and 1 in Base 2 equals 10_{2}. Subtraction uses similar additive inverse rules to other systems, so that subtraction means adding a negative number. More borrowing is used than in Base 10 subtraction.

Multiplication in Base 2 follows a similar repeated addition model as multiplication in decimal systems, and division in Base 2 follows a similar repeated subtraction model as division in decimal systems. Suppose that 110_{2 }is multiplied by 101_{2}. Partial products are used, so that the first row [110_{2 }times 1_{2}] is 110_{2}, the second row is [110_{2 }times 0] moved to the left one space, and the third row is [110_{2 }times 1_{2}] moved to the left 2 spaces. The entire product is 11110_{2}. It really helps to use graph paper or columnar ruled paper to keep the columns in line while adding or subtracting them.

Converting fractions or decimals into the binary Base 2 system presents a challenge, because many fractions and decimals that are exact in the decimal system have approximate values. The decimal 0.10 has an approximation in Base 2, and so do fractions such as ¼, 1/3, ½, and many others. Operations in Base 2 with fractions are approximate, called “floating point arithmetic.”

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]]>The concept of dividing a whole into parts and then dividing the parts into smaller entities is basic to mathematics. That model of division can be illustrated by manipulatives such as fraction bars and Cuisenaire rods. Equality and inequality of fractions can be demonstrated by finding their common denominators and comparing them. By comparing unequal fractions and finding more fractions between them, students can show the density of the number line.

A fraction is a part of a whole. Children can illustrate the concept by taking a whole circle, then cutting the circle in half, then cutting each half into halves to show fourths, then cutting each fourth into eighths, and so forth. Many different nations in ancient times, such as the Egyptians, Hindus, and Babylonians used fractions in different computations. The Arabs separated the numerator and a denominator by a bar between them.

Manipulatives such as fraction bars and Cuisenaire rods are special types of objects that can be used to illustrate the equality or inequality of fractions. Ten white cubes equal the same length as one orange rod, and so do five red rods. It can be shown that one red rod is the same length as two white cubes, in symbol terms 2/10 = 1/5. Similarly, 3 white cubes, or 3/10, is not the same size as 1 purple rod, or 1/3.

The best method to test whether fractions are equal or unequal is to find a common denominator for all the fractions in the group and then compare them. Suppose the fractions are 5/8 and 3/5. The fraction 5/8 is equal to 25/40, but the fraction 3/5 is equal to 24/40. They are not equal. Suppose the fractions are 3/5 and 6/10. Since 3/5 is equal to 6/10, the fractions are equal.

There is always another fraction that can be found between any two fractions on the number line. Take a closer look at the section between 0 and 1 on the number line. At first glance, it appears full. Halfway between 0 and 1 is ½, and halfway between that is ¼, and halfway between that is 1/8, then 1/16, then 1/32, and then there’s a point for ¾, and there’s also a point for 3/8, 5/8, and 7/8, 1/16, 3/16, 5/16, 7/16, 1/32, 3/32, and so on. Fractions are dense along the number line.

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]]>Some special factoring formulas include the difference of two squares, the sum of two cubes, and the difference of two cubes. If there are three terms or more in the polynomial, students can use strategies such as finding common factors and factoring by grouping.

The difference of two squares [(a + b)(a - b)] is a common pattern with binomials involving variables to the second power. However, the concept can also be applied to exponents higher than x^{2}. Any even power (such as x^{2}, x^{4}, x^{6}, and so on) can be factored into squares evenly. For example, 16x^{4} can be rewritten as (4x^{2})^{2} and 49y^{6} can be rewritten as (7y^{3})^{2}. The expression 16x^{4} – 49y^{6} is then factored as (4x^{2} – 7y^{3})(4x^{2} + 7y^{3}).

The product of (a + b)(a^{2} – ab + b^{2}) can be evaluated using FOIL as a^{3} –a^{2}b + ab^{2} +a^{2}b – ab^{2} + b^{3}. That simplifies to a^{3} + b^{3}. Suppose that the binomial that needs to be factored is 27x^{3} + 8. That expression will factor as (3x + 2)(9x^{2} – 6x + 4).

Figure 1: Factoring the sum of two cubes and the difference of two cubes.

The product of (a – b)(a^{2} + ab + b^{2}) can also be evaluated and simplified to a^{3} – b^{3}. The easiest way to remember the direction of the signs when factoring the sum or difference between two cubes is to use the acronym SOAP. The sign between the terms of the binomial factor is in the same direction in both the sum of the cubes and the (a + b) factor. (If the difference of cubes is the issue, the sign in a^{3} – b^{3} and a – b is negative.) The sign is opposite between the a^{2} term and the ab term, such that if it is the sum of cubes the sign between a^{2} and ab is negative, and if it is the difference in cubes, the sign between the a^{2} and ab term is positive. The sign between the ab term and the constant is always positive.

Figure 2: Using the acronym SOAP to remember the direction of the signs.

The first step in factoring a polynomial is always to factor out anything that is common to every term in the polynomial. Suppose that the polynomial to be factored is 3x^{2} + 6x + 9. The first step in factoring would be to remove the common factor of 3 from all the terms as 3(x^{2} + 2x + 3). Next, check to see if it follows any of the special factoring forms. It can be factored by grouping or another method.

Figure 3: Following the general steps to factor a polynomial.

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]]>Space figures exist in three dimensions. Some examples include spheres, cubes, tetrahedra, polyhedra, cylinders, cones, and other types of solids.

Plane figures such as polygons and curved figures are two-dimensional and flat. They have length and width. Space figures exist in three dimensions, adding depth. A sphere has a center point, and the myriad circles that exist around it form the sphere. A square in three dimensions is a cube.

A sphere has one fixed point at the center. When all points on the outside of a sphere are connected with its center, it becomes solid. Many real world objects are spherical or spheroid in shape, such as the earth and other planets, the sun and other stars, and bubbles. A round shape such as an oval can become a three-dimensional ovoid. Real-world examples include eggs or teardrops.

A tetrahedron is a solid with triangular faces. If the triangles that form a solid have any other polygon at the base, that solid is a pyramid. If other polygons are joined into a solid, that solid figure is called a polyhedron. If two faces of the polyhedron are congruent, such as a square at the top, and a congruent square at the bottom, the polyhedron is called a prism. Some polyhedra are regular, and others are irregular. Polyhedra have special properties; and their volume can be measured.

Cylinders are classic solids with two circular bases of the same size. If the bases are directly opposite each other, the figure is called a right cylinder. If the circular bases are not directly opposite, the figure is called an oblique cylinder. Cones are figures that rise from a circular base to a vertex. They may also be right or oblique. Classical solid figures include tetrahedra, cones, cubes, prisms, cylinders, and other polyhedra. However, other solid figures can be easily imagined as combinations of the classic shapes.

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]]>Plane figures in geometry include points, segments, lines, rays, and angles, polygons, curves, and circles. They appear on a flat plane.

Think of a pencil point, and make it smaller, until it is a no-dimensional, undefined, geometric concept that only exists in the minds of mathematicians. Segments are made up of points, and they consist of the beginning, the end and all points between the beginning and the end. They exist in two dimensions. Extend the segment to infinity in both directions, and it is a line. Start from only one point in any direction and extend it to infinity, and it is a ray. Rays are usually designated by the beginning point and the endpoint of a line segment. Two rays that meet at a common point, the vertex, are called an angle, and the angle can be measured with a protractor or duplicated using a drawing compass.

A three-sided polygon is called a triangle, because it consists of line segments arranged in three angles. Four-sided polygons may be squares, if all four sides are equal and each side meets at right angles. Other types of four-sided polygons include the rectangle, the rhombus, the parallelogram, and the trapezoid. Polygons are usually named for the number of sides they have; such as a pentagon with 5 sides, a hexagon with 6 sides, a heptagon with 7 sides, and octagon with 8 sides, and so on.

A set of points connected by a continuous line is called a curve. Curves can be open or closed. A circle is a type of closed curve that has a single center. Each point along the circumference of the circle is at the same distance from the center. The distance from the center to the circle itself is called the radius, and any circle measures 360^{o}.

Many other curved figures exist in geometry and in the real world, such as ellipses, arches, ovals, lenses, and crescents. Technically, polygons are also curves, because they can be formed by a continuous line, even if that line doesn’t appear curved.

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]]>If division were limited to single-digit divisors and math facts, it would be comparatively easy. Long division is more difficult to visualize and tie to real-world manipulative objects. Paradigms exist for long division, and there are tips and tricks for dealing with other problems.

Sometimes it is easier to tell if a dividend is evenly divisible by a divisor before working the problem out on paper. If a number in the units digit is divisible by 2 or 5, the entire number will be divisible by 2 or 5. For example, 385 is divisible by 5. Similarly, if the sum of the digits in a number is divisible by 3 or 9, the number is divisible by 3. If the sum of the last 2 digits is divisible by 4, the number will be divisible by 4. If a number is divisible by both 2 and 3, it will be evenly divisible by 6.

The steps for division include directions for repeated subtraction. This paradigm can be used for one-digit divisors as well as long division. It involves subtracting multiples of the divisor, adding the results up, and continuing until the leftover amount is less than the divisor. Suppose the number is 124/32. Then subtract 124 – 32 = 92 – 32 = 60 – 32 = 28. This method does not use place value, and makes it easier to see the concept.

One of the ways to look more carefully at division is to color-code both the dividend and the divisor using the color-coding that is available for chip-trading, such as red, thousands place, green, hundreds place, blue, tens place, and yellow, units place. Suppose the division problem is 4800/25. That would result in 4 red chips and 8 green chips. The 4 red chips cannot be evenly divided by 25, but they could be traded in for 40 green chips. The 40 green chips can be added to the 8 green chips that are already there to leave 48 green chips. The 48 green chips can then be divided by 25, which will leave 23 green chips. The 1 goes above the green. The 23 green chips become 230 blue chips. 225 of them (25*9) can be eliminated, and the 9 can go above the blue. The remaining 50 is evenly divisible by 25, so the solution is 100 + 90 + 2 = 192.

Normally, any number divided by itself is 1. What happens when zero is divided by itself? The answer is not 1, it is not determined. If any other number is divided by zero, the answer cannot be determined either. Dividing by zero is not allowed, because it doesn’t make any sense.

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]]>Division is the inverse of multiplication. If a times b equals c, and b is not equal to zero, then c divided by b equals a. Also, for any real number a not equal to zero, a times 1/a equals 1.

Division is one of the four basic arithmetic operations. It is usually the last one to be taught, because solving division problems can involve addition, subtraction, and multiplication. Suppose the problem is 36 divided by 4. The answer, 9, is a math fact, because 9 times 4 is 36. It can be solved by using the definition of division, as 36/4 =9. It can be solved by subtraction, as 36 – 4 = 32 – 4 = 28 – 4 = 24 – 4 = 20 – 4 = 16 – 4 = 12 – 4 = 8 – 4 = 4 – 4 = 0. The number 4 was subtracted 9 times from 36. To check division, students can multiply the quotient by the divisor, or add the divisor repeatedly.

Suppose you had 100 movie tickets and you wanted to give 4 free tickets to as many people as possible who were waiting in line outside a theatre. One way that could be done is to measure sets of 4 tickets and give them to the first people in line until there were no more left. In this case, 100 is the dividend, 4 is the divisor, and dividing 100/4, gives 25 groups of 4 tickets each. The first 25 people in line get 4 free movie tickets.

Similarly, suppose the 100 movie tickets were divided into 4 equal parts. How many tickets would be in each set? The quotient would be the same, 25, but this time the divisor 4 represents the number of equal parts. Even though the numbers are the same in both models, the concepts are a little different.

Many hands-on models for division exist, such as dividing beans or cubes into groups, or using the abacus to illustrate division by measurement. When students number off in groups of 4 to take part in activities or register for one of 5 home rooms, they illustrate the partitive model.

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]]>If two linear equations are solved, and their solution is graphed on a Cartesian coordinate plane, there are three possibilities. The lines may be the same, with all solutions in common. They may never intersect, with no solutions in common. They may intersect at only one point, with one solution in common.

The coordinate plane is also called the Cartesian plane, in honor of the great mathematician Rene Descartes. It is one of the most familiar graphs in mathematics, with an x-axis and a y-axis, stretching out to infinity in all directions. The point where the x-axis and y-axis cross is called the origin. Its coordinates are (0, 0). It is divided into four quadrants. They are usually called I, II, III, and IV. In Quadrant I, values of x and values of y are positive. In Quadrant II, values of x are negative and values of y are positive. In Quadrant III, values of x and values of y are negative. In Quadrant IV, values of x are positive and values of y are negative.

Figure 1: The coordinate plane.

If two or more equations have all solutions in common, the graph of their solutions will contain all solutions along the same line. In geometric terms, the points are collinear. It algebraic terms, the systems are dependent. Suppose the equations were x +y = 5 and 2x +2y =10. If x equals 1, then y equals 4; x equals 2, y equals 3, and so on. If the line for the equation x +y =5 is blue, and the graph of 2x + 2y = 10 is red, the lines will be superimposed, and the graphed collinear line is purple.

If two or more equations have no solutions in common, the graph of their solutions will contain all solutions along parallel lines. Suppose one equation is y = 3x + 2 and the other equation is y = 3x -1. The slope of each line is 2, so they have the same slope. The y-intercept of the first can be graphed vertical or perfectly horizontal have the same slope and different y-intercepts. Horizontal lines have a slope of zero by definition, but they each cross the y-axis at different points. Vertical lines have the same undefined slope, but they each cross the x-axis at different points.

Figure 2: When two linear equations have no solutions in common, their graphed lines are parallel.

Many systems of linear equations have a single solution that satisfies both equations. When those linear equations are graphed, they intersect at a single point. The linear equations may meet at many different angles, either acute or obtuse. When they meet at a 90 degree perpendicular angle, the values that solve each equation are in definite relationship to one another. Suppose that the equation for Line 1 has a slope of m_{1} and the equation for Line 2 has a slope of m_{2. }The product of m_{1} and m_{2} is -1. Similarly, if one equation can be graphed in a horizontal line parallel to the x axis, with slope 0, and the other equation in a vertical line parallel to the y axis, with undefined slope, the lines meet at one perpendicular point, a singular solution.

Figure 3: If a single solution satisfies both equations in a system, their graphed lines will meet at a single point.

Figure 4: The graphed lines will be perpendicular if the equations meet at a single point and the product of their slopes is -1.

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]]>Operations with sets include the union of sets, the intersection of sets, and complements of sets. These operations can be applied to solve survey problems using sets.

Some special types of sets include the null or empty set, the universal set, and proper and improper subsets. The null or empty set ⦰ has no elements. Its set notation is { }. In contrast, the universal set **U **contains all the members of every set in an operation. While the empty set is always the same, a set with no elements, the universal set changes with each problem. It is defined before any operations take place. Suppose that set F is the universal set for a problem. It contains the elements {2, 4, 6, 8, 10, 12}. Therefore, all operations on set F will contain only those elements. Proper subsets of Set F will include some of the elements of Set F, such as Set G {2, 4, 6} or Set H {8, 10}. In set language, the symbol shorthand is G ⊂ F. Improper subsets of Set F will contain all the members of Set F. Set I {12, 10, 8, 6, 4, 2} is an improper subset of Set F.

Figure 1: Set F is the universal set for this problem; Sets G, H, and M are proper subsets.

The union of sets is designated by the symbol ⋃, so if Set G {2,4,6} and Set H {8, 10} G ⋃ H = {2, 4, 6, 8, 10}. The intersection of sets consists of the elements that two or more sets have in common. Suppose that Set J {1, 2, 3,4, 5} and Set K {a, 3, d, 4, q}. The intersection of sets J⋂K is {3, 4}. Those are the only points common to sets J and K.

Figure 2: Venn diagrams as operational definitions of sets and operations.

The complement of a set is in relation to the universal set for that problem. Remember the universal set F with the elements {2, 4, 6, 8, 10, 12}? If Set O {6, 8, 10}, the complement of O (Ō), is {2, 4, 12}. If the universal set for a problem is** I**, the set of integers, and set L is {1, 3, 5, 7 …} the complement of L will be {0, 2, 4, 6 …}.

Figure 3: Some symbols and terminology of sets.

Suppose that 100 people are asked what types of shows they like to watch on TV. Suppose that 15 watch sports, 15 watch mysteries and true crime, 12 watch reality TV, 20 watch reality TV and mysteries, 28 watch sports and mysteries, and 3 watch sports, mysteries, and reality TV. How many people watch neither sports, mysteries, nor reality TV? This can be solved by using a Venn diagram and logic. Let Set S equal sports, Set M equal mysteries, and Set R equal reality TV. The intersection of S⋂M⋂R is 3, so the complement will be 15 -3 or 12. In other words, there are 12 others of the people who watch sports. Some of them watch only sports, some of them watch sports and mysteries, and some of them watch sports and reality TV. We already know that there are 15 people who watch mysteries, but some of them also watch sports, and some of them also watch reality TV. Similarly, the intersection of S⋂M⋂R is 3, so the complement will be 15 -3 or 12. There are 12 people who watch reality TV, but 3 of them also watch sports and mysteries, so the complement of 12-3 equals 9. The part of Set M that is not part of Set S is (15 – 3) or 12, and the part of Set M that is not part of set R is 12 -3 or 9. Finally, the part of Set R that is not part of Set S is 15 -9 or 6. 15 + 9 + 3 = 27. 100 – 27 = 73, or 73 people watch neither sports, mysteries, nor reality TV.

Figure 4: Venn diagram showing regions and relationships of 3 different sets.

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The dimensions of a matrix are important to its definition. In order for the multiplication of two matrices to be meaningful, each matrix must have certain dimensions. Multiplication of matrices is not commutative.

In order to multiply a matrix by a constant, every member of the matrix is multiplied by the constant. Suppose that matrix A consists of the members [wxyz]. Multiplying each member by a constant e means that the new matrix will consist of [ew ex ey ez]. Suppose Matrix B consists of [9 3 7 6]. Multiplying each member by 3 would lead to the new matrix [27 9 21 18].

Figure 1: Matrix A

Figure 2: In Matrix 1A, every element of Matrix A is multiplied by the constant e.

Figure 3: Matrix B is a 2 X 2 Matrix, in the same form as Matrix A, but the variables are replaced with real numbers.

Figure 4: In Matrix 1B, every element of Matrix B is multiplied by 3

The first step in multiplying matrices is to determine the dimensions of each matrix. If the number of columns of the first matrix is equal to the number of rows in the second matrix, multiplying the matrices is possible. Suppose Matrix D consists of the elements [1 3 5 7] and Matrix E consists of the elements [ 2 4 6 8 10 12]. Matrix D has the dimensions 2 X 2 and Matrix E has the dimensions 2 X 3. Matrix DE would also have the dimensions 2 X 3. It would be equal to multiplying the elements of Matrix D by the elements of Matrix E.

Figure 5: Matrix D

Figure 6: Matrix E

In the next step, multiply each element of the row in the first matrix by each element of the column of the second matrix, and add those products together to form the new elements of the product matrix. Using the example of Matrix DE, the first row of the new matrix would equal (1∙2 + 3∙8) (1∙4 + 3∙10) (1∙6 + 3∙12), or (2 + 24) (4 + 30) (6 + 36). The second row of the new matrix DE would equal each element of the second row of the first matrix multiplied by each element of the column of the second matrix or (5∙2 + 7∙8) (5∙4 + 7∙10) (5∙6 + 7∙12) or (10 + 56) (20 + 70) (30 + 84). The new matrix would consist of the elements [26 34 42 66 90 114] in a 2 x 3 matrix. .

Figure 7: Matrix DE

Suppose Matrix F consisted of the elements [1 2 3 4] in a 2 x 2 matrix and Matrix G consisted of the elements [ 0 1 3 5 7 9] in a 2 x 3 matrix.. The new Matrix FG would consist of the elements [10 15 21 20 31 44] in a 2 x 3 matrix. Matrix GF would be a 2 X3 matrix X a 2 X2 matrix, which would be undefined, as the number of columns in G (3) is not equal to the number of rows in F(2).

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