Number and Quantity

Number and Quantity
Created by: Chelsea, Ellen, Michael, and Mitch
Sections:
  • The Real Number System
  • Quantities
  • The Complex Number System
  • Vector and Matrix Quantities
History of Number:
Our modern number system, which consists of a combination of ten symbols (1, 2, 3, 4, 5, 6, 7, 8, 9, and 0) were introduced in Europe by Leonardo Pisano, an Italian Mathematician. Many number systems preceded the system we currently have. The first Western use of the numerals, without zero, was recorded in the 5th century by Beothius, a Roman writer.

Teaching Goals:

Number: TLW extend their understand of numbers and number systems as they progress through their math classes.
  1. "Counting numbers" - 1, 2, 3, ...
  2. "Zero" - having nothing present
  3. Whole Numbers (counting numbers and zero)
  4. Negative numbers
  5. Rational numbers - introduce fractions, first pictorally

Quantity: TLW understand and be able to use units to solve problems
  1. examples of quantity: length, area, volume, average, income. etc.

Definitions:
Number: the sum, total, or count of a collection of units.
Quantities:
  1. the property of magnitude involving comparability with other magnitudes.
  2. something having magnitude, or size, extent, amount, or the like.
  3. magnitude, size, volume, area, or length.

Variable: a number or quantity that may vary or change

Constant: a number or quantity that remains unchanged

Vector and Matrix Quantities
Common Core Standards:
Vector:
  1. a mathematical entity that has both magnitude (can be zero) and direction
  2. an element of a vector space

Picture_2.png
Matrix:
  1. Matrix notation can be used to represent a system of linear equations. It is more compact, showing only the coefficients in a rectangular array.



The Real Number System:
Common Core State Standards:
  1. Extend the properties of exponents to rational exponents.
  2. Use properties of rational and irrational numbers
    1. The sum or products of two rational numbers is rational
    2. The sum of a rational number and an irrational number is irrational
    3. The product of a nonzero rational number and an irrational number is irrational

A rational number is a number that can be written in the form of a fraction. For example, one basic idea is the fraction 1/2. In this case, 1 is the numerator and two is the denominator. All whole numbers can be written as fractions. For example, the whole number 5 can be written as 5/1, where 5 is the numerator and 1 is the denominator. The root of a number can not always be written as a factional. At the same time, roots can be rewritten as fractional exponents. For example, the square root of 2 can be written as 2^(1/2) and the cube root can be written as 2^(1/3). Essentially, to eliminate the radical, the root is written as a factional exponent. The number inside the radical remains constant and the root is written as the inverse in the exponent. Thus, the sqrt2 is equal to 2^(1/2).

The sum or products od two rational number is irrational. Integers are closed under addition and subtraction. That is, an integer plus/times an integer is another integer. Rationals can be written as fractions. When we add fractions, we find like denominators and add the numerators together. When we multiply fractions, we multply the numerators together and the denominators together. Therefore, there will remain an integer in the numerator as well as in the denominator. Thus, we have an ineger divided by an integer, which is a fraction. A fraction is a rational number.

Adding a rational number to an irrational number is still an irrational number. For example, the number pi is irrational since it can not be written as a fraction. It is a number with an infinatly long decimal. Any number added or subtracted from infinity is still infinitely long. Since it is still infinitely long, it can not be written as a faction.

The product of a nonzero rational number and an irrational number is irrational. An irrational number can be written as a decimal that is infinitely long. A rational number multiplied by infinity will will be infinitely long. Infinity is not a number, but an idea of a number that never ends. This idea can not be contain in a single numerical value. For that reason, it continues to be infinitely long and is irrational.


Quantities:
Common Core State Standards:
  1. Use units as a way to understand problems and to guide the solution to multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
  2. Define appropriate quantities for the purpose of descriptive modeling.
  3. Choose a level of accuracy appropriate to limitations on measurements when reporting quantities.

Quantities are the way in which we verbally express and describe algebraic reasoning. An emphasis on words introduces a new level of understanding of mathematics. First, we begin by identifying basic quantities.
  1. height
  2. depth
  3. area
  4. length
  5. width
  6. volume
  7. time
  8. speed

Once we have some idea of basic quantities we look at how we use words in our everyday language to describe these without knowing the exact quantity. For example as we watch the flight of a plane we can say that its height above the ground at take off is 0. This height increases until it reaches its maximum or cruising height. Once this height is obtained it will be relatively level before finally decreasing as the plane approaches landing at which point the height will be back to 0.
Another example is keeping track of the amount of people in a room. As people enter the room the amount increases, the reverse happens as people leave the room the amount decreases. If no one enters or leaves the room and or if the amount that enters is approximately the same as the number that leaves the amount of people in the room will remain approximately equal.

Next we look at how we use specific words such as variables and constants to describe mathematical quantities. We say a quantity is variable when it can vary or change. For example, the time of day is constantly changing and therefore time is variable. Likewise, the position of the sun in the sky is constantly changing and also variable. The opposite of this idea would be a constant quantity. We refer to a quantity as constant when the value is unchanging or remains the same. A car that is stopped at a stop light has a speed of 0 and will until it is no longer stopped. This means that the speed for this time is constant.
It is also important to note that whether a quantity is variable or constant it can also be known or unknown. A quantity maybe known to some and unknown to others. For example the architect of a skyscraper may know the exact height of the building while you may not therefore the height is known to the architect and unknown to you.



The Complex Number System:
Common Core State Standards

A complex number is an expression in the form a+bi where a and b real numbers. The symbol i is defined as


I.jpg=i2.png
The imaginary number i was created do to the fact that equations such as x^2+1=0 were thought to be impossible to solve. There are surprisingly many uses for complex numbers. Engineers use it to study stresses on beams and to study resonance. Complex numbers help us study the flow of fluid around objects, such as water around pipes. They are also used in electric circuits, and help in transmitting radio waves. Finally, the complex number system is used in studying finite series, and each polynomial equation has a solution if complex numbers are used.


  1. Perform arithmetic operations with complex numbers

When adding and subtracting complex numbers we need to know that a is the real part of the complex number, and b is the complex part of the complex number. To add/subtract two complex numbers, add/subtract the real part of the first number with the real part of the second number. Then we want to add/subtract the imaginary part of the first number with the imaginary part of the second number. For example:

(-4-2i)+(6-4i) (5-4i)-(2+3i)
(-4+6)+(-2-4)i (5+2)+(-4-3)i
2-6i 7-7i
When multiplying two complex numbers, set up the complex numbers like two binomials and use the distributive property for binomials (FOIL method). Then use the fact that i^2=-1, and combine line terms. For example:

(-3+2i)(4+i)
-12-3i+8i+2i^2
-12-3i+8i+2(-1)
-14+5i

Before dividing complex numbers we must understand the complex conjugate. If we have the complex number as a+bi then it's complex conjugate is a-bi. To find the complex conjugate we are simply negating the sign in front of the complex part of the complex number. For example:




Complex Number
Conjugate Number
2-4i
2+4i
-3+2i
-3-2i


To divide two complex numbers, arrange the complex numbers into a fraction with the divisor as the numerator and the dividend as the denominator. Next, multiply the top and bottom of the fraction by the complex conjugate of the denominator, and combine like terms. For example:



Dividing_Complex_Numbers.jpg



Information from: http://bit.ly/rPJqbC
More examples can be found at: http://bit.ly/vsUfFG

2. Represent complex numbers and their operations on the complex plane

We can visualize complex numbers by associating them with points on a plane. We can do this by letting the complex number a+bi correspond to the point (a,b):

cplane.gif
We can see that the complex number -3+5i is at he point (-3,5). However, we must remember that these are complex numbers so the complex plane without any points looks like:


complex_plane.gif

So if we are looking a complex number, a+bi, our point on the complex plane would be (a,bi).

3. Use complex numbers in polynomial identities and equation

The many formulas and properties for polynomials with real coefficients also hold for polynomials with complex coefficients. An example of solving a polynomial with complex coefficients is when trying to solve the equations x^2+25=0. There are many ways to solve this, but I would try and use the quadratic formula or you can see an example of using factoring below:


03-complex-numbers-10.gif

  1. Represent and model with vector quantities
    1. Vectors have both direction and magnitude, and can be represented graphically as a directed line segment. Students should also know the symbols used for vectors and vector magnitude.
    2. Students should be able to solve problems that model information using vectors, such as velocity problems.
  2. Perform operations on vectors
    1. Add and subtract vectors.
    2. Multiply a vector by a scalar.
  3. Perform operations on matrices and use matrices in applications.
    1. Use matrices to represent data.
    2. Multiply matrices by a scalar and interpret the results in terms of the represented data of the problem.
    3. Add, subtract, and multiply matrices.
    4. Understand that the zero and identity matrices play similar roles to addition and multiplication to that of 0 and 1 in the real numbers.

Example Problems:

2.1 Adding vectors component-wise and using the parallelogram rule.
A = <1,5>
B = <3,1>
A + B = <1,5> + <3,1> = <4,6>

Below, using the parallelogram rule:

Picture_1.png

If you move either A or B so that the beginning is lined up with the end of the other, you will end up at the same place as the vector A + B. That is, the end point of the vector D is the fourth vertex of the parallelogram whose other vertices are A, 0, and B.

2.2 Multiplying Vectors by a Scalar:

c<a, b> = <ca, cb>, where c is a real number and <a, b> is a vector

3.4 Zero and Identity Matrices
Zero Matrix: (0) the zero matrix is an n x m matrix where n, m are integers such that all entries are zero.
Identity Matrix: (I) The simplest n x n nontrivial matrix such I (x) = x for all vectors x. Shown below is a 3 x 3 identity matrix.

wolframalpha.gif

How are the roles of zero matrices and identity matrices similar to 0 and 1 in the real numbers relating to addition and multiplication?
This is a good opportunity to student exploration since they already know how 0 and 1 behave in the real numbers in relation to addition and multiplication.

Examples of multiplying by the 2 x 2 identity matrix:
wolframalpha-1.gif
wolframalpha-3.gif

Sources:
http://www.archimedes-lab.org/numeral2.html