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- Inclusive and Exclusive Definitions [04/05/2001]
Are squares rectangles? Are rectangles squares?
- Inclusive Definitions: Trapezoids [11/04/2004]
As far as I know, a trapezoid is defined as a quadrilateral with exactly one set of parallel sides. However, a very highly regarded educator and textbook author recently argued that this definition is incorrect. His definition of a trapezoid is that it is a quadrilateral that has at least one pair of parallel sides. A square, therefore, would be considered a trapezoid. Is he correct or are thousands of books going to be published with the wrong definition?
- Inclusive vs. Exclusive Definitions [01/24/2002]
My geometry teacher says that a square is not also a rhombus, a
rectangle, and a parallelogram. Please help!
- Incommensurable Numbers [07/25/2001]
What is an incommensurable number?
- (In)consistent and (In)dependent Systems [01/02/2007]
Systems of equations are classified using the terms consistent,
inconsistent, dependent, and independent. I'm curious how those terms
were chosen and how they fit the solutions of the systems.
- Inconsistently Improper? [05/07/2017]
A teen tries to reconcile why we call an algebraic fraction improper if it evaluates to a proper
fraction. Doctor Peterson clarifies how the scope of the terms "proper" and "improper" differs when applied to arithmetic versus algebra.
- Interpolation and Extrapolation [08/10/1999]
Can you give me a step-by-step procedure for finding missing values based
on interpolation and extrapolation?
- Interpreting Expressions Involving Cosine Squared [03/16/2007]
I don't understand the difference between cos(x)^2, cos^2(x) and (cos
(x))^2. Are they all same?
- Introduction to Quadrants in the Coordinate Plane [02/05/2004]
What is a quadrant? How do you find it? I had to identify 9 ordered
pairs on a graph. Now I need to also name the quadrant each one is in.
- Inverse of a Function vs. Inverse Proportionality [11/20/2002]
Finding the inverse of a function and graphing yields a graph that has
been reflected in the line y = x relative to the function. Inverse
proportionality, however, yields a reciprocal relation graphically.
Why do these two things have similar names yet mean different things?
- Inverses of Functions, and Inverse Functions [02/12/2014]
A student's textbooks give her conflicting definitions about functions and their
inverses. Doctor Peterson makes sense of the disagreements by providing the bigger
- Is a Circle a Polygon? [03/07/1999]
A circle is not a polygon.
- Is an Equilateral Triangle Also Isosceles? [02/19/2008]
Can equilateral triangles also be classified as isosceles? In other
words, does an isosceles triangle have exactly two equal sides or at
least two equal sides?
- Is a Sphere 2-D or 3-D? [8/8/1996]
Is a sphere a two- or a three-dimensional object?
- Is Kite the True Name? [03/29/2002]
Is kite the true math name for this shape, or is there another?
- Isomorphisms [08/16/1999]
Can you give me an explanation and a nice example of isomorphism?
- Is Zero Considered a Pure Imaginary Number (as 0i)? [12/02/2003]
In the complex plane, zero (0 + 0i) is on both the real and pure
imaginary axes. Is 0 therefore a pure imaginary number as well as a
- Is Zero Really a Number or Just a Concept? [09/15/2005]
I've been taught that infinity is just a concept, not an actual
number. I'm wondering if the same thing can be said of zero?
- Josephus Problem [04/18/2003]
Every other person at a table is eliminated until there is only one
person left. Who is the survivor?
- The Königsberg Bridge [5/20/1996]
Do you have information on Konigsberg's bridge?
- Lagrange Multipliers [01/08/1998]
I have a problem with Lagrange Multipliers - can you help?
- Larger Than and As Large As [08/01/1999]
If the percussion section has 6 people, does the expression "the brass
section is three times larger than the percussion section" mean that the
brass section is 3*6 = 18 people or 4*6 = 24 people?
- Latin Origins of Trig Functions [11/20/1998]
What are the etymologies of the six trigonometric functions: sine,
cosine, tangent, cosecant, secant, and cotangent?
- Laws of Arithmetic [02/12/1999]
The Distributive Law, Associative Law, Identity, and Commutative
- Leg of a Triangle [04/02/2002]
I need to know where the name "leg" of a triangle comes from, or what its
- Length of a Light-Year [1/31/1996]
How far is a light year in miles or kilometers?
- Limit of x sin(1/x) [04/23/2002]
I assumed from the graph that the function had a limit at x=0 of 0,
but since it involves sin(1/0) I can not prove this using the basic
trigonometric limits (sin x/x and (1-cos x)/x), L'Hopital's
rule, or by rearranging the equation. Can you help?
- Linear Algebra - Definition of Orthonormal [10/28/1997]
Show that the transition matrix P from one orthonormal basis to another
is unitary, that is, P*P = I.
- Linear and Board Feet [01/04/1999]
Can you explain the terms linear foot and board foot as they are used in
the lumber industry?
- Linear Footage [04/20/2002]
There is a fence I want to buy, but the ad says '4 foot tall, 50
linear feet. What is a linear foot compared to a regular foot?
- Line of Best Fit [8/1/1996]
Are a best-fit line, a line of best fit, and a fitted line synonymous?
- Line of Best Fit, and Flight [02/04/2015]
What do "extrapolation" and "interpolation" mean? What are their purpose? Doctor
Peterson defines the terms, then gives real-world examples based on tracking the
flight of his son's recent airplane trip.
- Line or Ray Longer? [12/11/2001]
Which is longer, a ray or a line?
- Locus [05/03/1999]
What is a locus?
- Locus Locution [11/10/2016]
A teen struggles for the right way to phrase a term of plane geometry. With citations
and examples, Doctor Peterson outlines how the context of the term has changed over
- Marginal Meanings [12/08/2016]
Having calculated step-wise differences in output of the cost function, an adult wonders why those values disagree with output of the derivative of the same function. Acknowledging the ambiguities, Doctor Peterson makes sense of the approximations.
- Mass and Weight [08/16/1997]
Why are mass and weight both measured with the same unit? If weight is
different on the moon and mass is not, what is the difference?
- Mass versus Weight [09/09/99]
Do mass and weight refer to the same thing on earth? Is mass equal to
- Mass vs. Weight [01/22/1999]
Can you give me a definition of mass and how it differs from weight?
- Mathematical Induction [09/07/1998]
What is mathematical induction? Can you give an example of the ideas of