Number and Quantity

The One Thing

Number and quantity extends ordinary arithmetic from real numbers to complex numbers, vectors, and matrices. Each system supports specific operations whose validity depends on definitions, algebraic properties, and—in the case of matrices—compatible dimensions.

Definitions and Results

• Real number: A number that can be represented on a number line, including rational and irrational numbers.
• Complex number: A number written as $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.
• Imaginary unit: The number $i$ defined by $i^2=-1$, so $i=\sqrt{-1}$.
• Complex conjugate: For $a+bi$, the conjugate is $a-bi$; multiplying conjugates gives

$$(a+bi)(a-bi)=a^2+b^2.$$

• Absolute value of a complex number: The modulus of $a+bi$ is

$$|a+bi|=\sqrt{a^2+b^2},$$ representing its distance from the origin in the complex plane.

• Fundamental Theorem of Algebra: Every polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots when multiplicity is counted.
• Vector: A quantity represented by an ordered list of components, often describing both magnitude and direction. In two dimensions, a vector may be written as $⟨a,b⟩$ or as the column matrix

$$\left[\begin{array}{c}a\\ b\end{array}\right].$$

• Vector addition: Vectors are added component by component:

$$(a,b)+(c,d)=(a+c,b+d).$$

• Scalar multiplication: Every component is multiplied by the same scalar:

$$k(a,b)=(ka,kb).$$

• Magnitude of a vector: For $v=(a,b)$,

$$\Vert v\Vert =\sqrt{a^2+b^2}.$$ In three dimensions, $$\Vert v\Vert =\sqrt{a^2+b^2+c^2}.$$

• Matrix: A rectangular arrangement of numbers organized into rows and columns.
• Matrix dimensions: Dimensions are stated as rows by columns, such as a $2$ by $3$ matrix.
• Matrix addition: Matrices of the same dimensions are added by adding corresponding entries.
• Matrix multiplication: If $A$ is an $m$ by $n$ matrix and $B$ is an $n$ by $p$ matrix, then $AB$ is defined and has dimensions $m$ by $p$. Each entry is calculated using row-by-column products and sums.
• Identity matrix: A square matrix with $1$s on the main diagonal and $0$s elsewhere. Multiplication by the identity matrix leaves a compatible matrix unchanged.
• Matrix transformation: A matrix can act on a vector to rotate, reflect, stretch, shrink, or otherwise transform it.
• Closure: A set is closed under an operation when applying that operation to members of the set produces another member of the same set.
• Associative property: Grouping can change without changing the result:

$$(a+b)+c=a+(b+c),$$ and similarly for multiplication when defined.

• Distributive property: Multiplication distributes over addition:

$$a(b+c)=ab+ac.$$

The number systems are nested: $$\text{natural numbers}\subseteq \text{whole numbers}\subseteq \text{integers}\subseteq \text{rational numbers}\subseteq \text{real numbers}\subseteq \text{complex numbers}.$$

Worked Methods

1. Working with powers of $i$

Use the repeating cycle of four powers: $$i^1=i,i^2=-1,i^3=-i,i^4=1.$$

The pattern repeats every four powers, so higher powers can be reduced according to their remainder when divided by $4$.

2. Adding and subtracting complex numbers

Combine real parts separately and imaginary parts separately: $$(a+bi)+(c+di)=(a+c)+(b+d)i.$$

Subtraction follows the same componentwise procedure.

3. Multiplying complex numbers

Apply the distributive property and replace $i^2$ with $-1$: $$(a+bi)(c+di)=ac+adi+bci+bd{i}^2=(ac-bd)+(ad+bc)i.$$

4. Dividing complex numbers

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. This removes the imaginary part from the denominator because $$(c+di)(c-di)=c^2+d^2.$$

5. Solving a quadratic equation

For $$a{x}^2+bx+c=0,a\ne 0,$$ use the quadratic formula: $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

The discriminant is $$b^2-4ac.$$

Its value determines the roots:

• If $b^2-4ac>0$, there are two real roots.
• If $b^2-4ac=0$, there is one repeated real root.
• If $b^2-4ac<0$, there is a pair of complex conjugate roots.

The Fundamental Theorem of Algebra extends this result: every degree-$n$ polynomial with complex coefficients has exactly $n$ complex roots when multiplicity is counted.

6. Adding vectors

Add corresponding components: $$(a,b)+(c,d)=(a+c,b+d).$$

The same rule applies when vectors are written using angle brackets or column matrices.

7. Multiplying a vector by a scalar

Multiply every component by the scalar: $$k(a,b)=(ka,kb).$$

8. Finding the magnitude and a unit vector

For $v=⟨a,b⟩$, calculate $$\Vert v\Vert =\sqrt{a^2+b^2}.$$

For a three-dimensional vector, use $$\Vert v\Vert =\sqrt{a^2+b^2+c^2}.$$

A unit vector has magnitude $1$. For a nonzero vector $v$, obtain its unit vector by dividing by its magnitude: $$\frac{v}{\Vert v\Vert }.$$

9. Calculating a dot product

For $$u=⟨a,b⟩,v=⟨c,d⟩,$$ calculate $$u\cdot v=ac+bd.$$

The dot product can be used to determine angles and to test perpendicularity. Two vectors are perpendicular when their dot product is $0$.

10. Adding and subtracting matrices

First check that the matrices have the same dimensions. Then add or subtract corresponding entries. Matrix addition and subtraction are not defined for matrices with different dimensions.

11. Multiplying matrices

For matrices $A$ and $B$:

• Check that the number of columns in $A$ equals the number of rows in $B$.
• Determine the dimensions of the product from the outer dimensions.
• Calculate each entry by multiplying a row of $A$ by a column of $B$ and summing the products.

If $A$ is $m$ by $n$ and $B$ is $n$ by $p$, then $AB$ is $m$ by $p$.

Matrix multiplication is generally not commutative: $$AB\ne BA.$$ Moreover, one product may be defined while the other is not.

12. Using matrices for systems and transformations

A system of linear equations can be represented by a matrix or an augmented matrix. Row operations can then be used to find solutions.

A matrix can also act on a vector as a transformation, including rotation, reflection, stretching, shrinking, or other transformations in the coordinate plane. Matrix and vector models are used in computer graphics, physics, engineering, economics, data organization, and transformations in the coordinate plane.

Where It Goes Wrong

• Treating $i^2$ as $1$ rather than $-1$, or failing to use the four-power cycle $i,i^2,i^3,i^4$.
• Combining complex numbers incorrectly instead of combining real parts and imaginary parts separately.
• Dividing complex numbers without multiplying the numerator and denominator by the conjugate of the denominator.
• Applying the quadratic formula without checking the condition $a\ne 0$, or misinterpreting the discriminant.
• Adding or subtracting matrices with different dimensions.
• Multiplying matrices in the wrong order, ignoring the row-column compatibility condition, or assuming that $AB=BA$.

What Gets Asked

This material supports questions requiring students to:

• Classify numbers within the nested systems from natural numbers through complex numbers.
• Simplify powers of $i$, and add, subtract, multiply, or divide complex numbers.
• Use complex conjugates and calculate the modulus of a complex number.
• Solve quadratic equations using the quadratic formula and interpret the discriminant.
• State or apply the Fundamental Theorem of Algebra.
• Add vectors, multiply vectors by scalars, calculate vector magnitudes, find unit vectors, and compute dot products.
• Determine whether vectors are perpendicular.
• Identify matrix dimensions and test whether matrix addition, subtraction, or multiplication is defined.
• Perform matrix addition and row-by-column multiplication.
• Explain why matrix multiplication is not generally commutative.
• Use the identity matrix, row operations, matrices, and vectors to represent systems and transformations.
• Apply closure, associativity, and distributivity to determine which algebraic manipulations are valid.