Sandwich rule

○○○ bijection ⪢⪢
A function f:A￫B where A and B are intervals is a bijection if, for every b ∈ B, there exists a unique a ∈ A with f(a) = b.
5079#9535 SBLNGS CHLDRN 9535

○○○ bounded above ⪢⪢
A subset S of R is said to be bounded above if there is some real number M such that x≤M for all x∈S. This M is called an upper bound for S.
5079#5041 SBLNGS CHLDRN 5041

○○○ bounded below ⪢⪢
A subset S of R is said to be bounded below if there is some real number m such that x≥m for all x∈S. This m is called a lower bound for S.
5079#5066 SBLNGS CHLDRN 5066

○○○ calculus ⪢⪢
5079#3789 SBLNGS CHLDRN 3789

◌◌◌ Calculus
5079#5053 SBLNGS CHLDRN 5053

○○○ cardinal number ⪢⪢
A _____ says how many of something there are. ￫ nominal, ordinal
5079#4825 SBLNGS CHLDRN 4825

○○○ combinatorics ⪢⪢
5079#4988 SBLNGS CHLDRN 4988

◎◎◎ complement ⪢⪢
When two angles are complementary, we say that one angle is the _____ of the other. #math
5079#9593 SBLNGS CHLDRN 9593

◎◎◎ complementary ⪢⪢
If two angles are _____ , then the sum of their measures is 90 degrees. ￫ measure #math #bookmark
5079#9684 SBLNGS CHLDRN 9684

○○○ complex number ⪢⪢
A _____ is a number that can be expressed in the form a + bi where a and b are real numbers and i is the square root of -1.
5079#9607 SBLNGS CHLDRN 9607

○○○ Composite rule ⪢⪢
If ⎾f(x)⏐x￫a⏌= L and g is a function that is continuous in some neighborhood of x=L, then ⎾(g∘f)(x)⏐x￫a⏌= g(L).
5079#6085 SBLNGS CHLDRN 6085

○○○ composite number ⪢⪢
It is a _____ when it can be divided evenly by numbers other than 1 or itself.
5079#4824 SBLNGS CHLDRN 4824

○○○ Composite rule ⪢⪢
If ⎾lim f(x)⏐x￫a⏌= L and g is a function that is continuous in some neighborhood of x=L, then ⎾lim (g∘f)(x)⏐x￫a⏌= g(L).
5079#5000 SBLNGS CHLDRN 5000

○○○ Composite rule ⪢⪢
Let f and g be continuous at c and f(c) respectively, such that g∘f is defined. Then g∘f is continuous at c.
5079#9097 SBLNGS CHLDRN 9097

○○○ continuity ⪢⪢
5079#5080 SBLNGS CHLDRN 5080

• Cross Product
Given two nonzero vectors in two or three dimensions, their cross product is a vector with magnitude equal to the product of the magnitudes of the vectors times the sine of the angle between the vectors and direction perpendicular to the vectors.
5079#5291 SBLNGS CHLDRN 5291

○○○ Definition ⎾lim f(x)⏐x￫∞⏌= L ⪢⪢
f(x) tends to L as x tends to infinity, if for every ε>0 there exists a real number X such that x>X ⇒ ⎹f(x)−L⎸<ε.
5079#4818 SBLNGS CHLDRN 4818

○○○ Definition ⎾lim f(x)︱x￫a⏌= L ⪢⪢
f(x) tends to L as x tends to a, if and only if for every ε>0 there exists a δ>0 such that 0<⎹x−a⎸<δ ⇒ ⎹f(x)−L⎸<ε.
5079#4819 SBLNGS CHLDRN 4819

○○○ denominator ⪢⪢
5079#4826 SBLNGS CHLDRN 4826

◎◎◎ digit ⪢⪢
A _____ is a single symbol used to make numerals.
5079#4827 SBLNGS CHLDRN 4827

• digits, numerals and numbers
A number is a count or measurement. It is really an idea in our minds. ○ count: the total determined by counting ○ measurement: the size, length or amount of something, as established by measuring
5079#9538 SBLNGS CHLDRN 9538

◎◎◎ dimension ⪢⪢
The number of _____s is how many values are needed to locate points on a shape.
5079#4828 SBLNGS CHLDRN 4828

◌◌◌ e
5079#5108 SBLNGS CHLDRN 5108

◌◌◌ function
5079#5109 SBLNGS CHLDRN 5109

○○○ gauge ⪢⪢
But for now you can just think of a gauge as meaning a choice of coordinate system.
5079#20854 SBLNGS CHLDRN 20854

○○○ geometry ⪢⪢
5079#3788 SBLNGS CHLDRN 3788

○○○ Golden Section ⪢⪢
A length is devided into two parts in such a way that the smaller part is to the larger part in the same proportion as the larger one is to the whole. #math
5079#1780 SBLNGS CHLDRN 1780

• Greek alphabet
Α α Β β Γ γ Δ δ Ε ε Ζ ζ Η η Θ θ Ι ι Κ κ Λ λ Μ μ Ν ν Ξ ξ Ο ο Π π Ρ ρ ϱ Σ σ/ς Τ τ Υ υ Φ φ Χ χ Ψ ψ Ω ω
5079#5203 SBLNGS CHLDRN 5203

◌◌◌ Integration
With the sum of f of x i times delta x from i equals one to infinity in the limit of n approaching infinity.
5079#3658 SBLNGS CHLDRN 3658

○○○ Inverse rule ⪢⪢
Suppose that f:A￫B is a bisection where A and B are intervals. If f is continuous on A, then f˂ is continuous on B.
5079#9603 SBLNGS CHLDRN 9603

• irrational number
Irrational numbers can't be expressed as the ratio of two integers. #math
5079#9591 SBLNGS CHLDRN 9591

○○○ Isometric Transformations ⪢⪢
_____ are types of transformations where the angles, sides, side measurements and the size of the figure remain the same.
5079#3849 SBLNGS CHLDRN 3849

○○○ Left and Right Theorem ⪢⪢
⎾lim f(x)⏐x￫a⏌= L ⇔ ⎾lim f(x)⏐x￫a⁻⏌=⎾lim f(x)⏐x￫a⁺⏌= L
5079#4993 SBLNGS CHLDRN 4993

○○○ Left and Right Theorem ⪢⪢
f is continuous at c if and only if f is both left-continuous and right-continuous at c.
5079#4994 SBLNGS CHLDRN 4994

◌◌◌ limit
5079#5110 SBLNGS CHLDRN 5110

○○○ limit⎾f(x)⏐x￫∞⏌= L ⪢⪢
Let f:A￫R be a function whose domain A contains the interval [a,∞) for some fixed real value of a. Then f(x) tends to L as x tends to infinity, if for every ε>0 there exists a real number X such that x>X ⇒ ⎹f(x)−L⎸<ε.
5079#3473 SBLNGS CHLDRN 3473

○○○ limit⎾f(x)︱x￫a⏌= L ⪢⪢
f(x) tends to L as x tends to a, if and only if for every ε>0 there exists a δ>0 such that 0<⎹x−a⎸<δ ⇒ ⎹f(x)−L⎸<ε.
5079#5112 SBLNGS CHLDRN 5112

• Linear systems
A collection of 2 or more linear equations
5079#3810 SBLNGS CHLDRN 3810

○○○ linearize ⪢⪢
5079#20853 SBLNGS CHLDRN 20853

○○○ log/root ⪢⪢
Root and log are the inverse operations of exponentiation. b˄p = y ┆ y˅p = b ┆ y⍻b = p
5079#5054 SBLNGS CHLDRN 5054

◌◌◌ loop
5079#16024 SBLNGS CHLDRN 16024

○○○ Lorenz Gauge ⪢⪢
5079#20852 SBLNGS CHLDRN 20852

○○○ mathematical logic ⪢⪢
5079#4991 SBLNGS CHLDRN 4991

◌◌◌ mathematics
5079#3736 SBLNGS CHLDRN 3736

○○○ mold ⪢⪢
A sphere and a cube are topologically the same thing since you can just kind of mold one into the other. #math
5079#16021 SBLNGS CHLDRN 16021

○○○ nominal number ⪢⪢
A _____ is a number used like a name, not as an actual value or position. ￫ cardinal, ordinal
5079#4829 SBLNGS CHLDRN 4829

◎◎◎ number ⪢⪢
Digits make up numerals and numerals stand for an idea of a _____ just like letters make up words and words stand for ideas.
5079#4832 SBLNGS CHLDRN 4832

◎◎◎ number ⪢⪢
A _____ is a count or measurement. It is really an idea in our minds.
5079#4833 SBLNGS CHLDRN 4833

○○○ number ⪢⪢
Here are some ways to show the number 4. We can even tap the ground 4 times.
5079#4938 SBLNGS CHLDRN 4938

• Number Systems
5079#6086 SBLNGS CHLDRN 6086

○○○ number theory ⪢⪢
5079#4985 SBLNGS CHLDRN 4985

○○○ ordinal number ⪢⪢
￫ cardinal, nominal
5079#4830 SBLNGS CHLDRN 4830

○○○ perimeter ⪢⪢
The perimeter is the distance around the edge of _____ triangle. #math
5079#18161 SBLNGS CHLDRN 18161

○○○ power rule ⪢⪢
(x˄n)ᐁx = n·x˄(n−1) #math #bookmark
5079#9686 SBLNGS CHLDRN 9686

○○○ prime number ⪢⪢
It is a _____ when it can't be divided evenly by any number except 1 or itself.
5079#9581 SBLNGS CHLDRN 9581

• prime number theorem
○ π(n) is asymptotically equivalent to x/log x. ○ Of the first n integers, roughly 1/log n of them would be prime. #math
5079#9583 SBLNGS CHLDRN 9583

○○○ Pythagorean Theorem ⪢⪢
The _____ says that for any right triangle with sides a, b and c, we have a²+b²=c².
5079#28252 SBLNGS CHLDRN 28252

◌◌◌ Quadrilateral
5079#5167 SBLNGS CHLDRN 5167

○○○ quadrilateral ⪢⪢
5079#11766 SBLNGS CHLDRN 11766

○○○ rectangle ⪢⪢
5079#20604 SBLNGS CHLDRN 20604

○○○ rhombus ⪢⪢
If all four sides are the same length, we call the shape a _____. #math #bookmark
5079#20603 SBLNGS CHLDRN 20603

• Roman Numerals
The Romans showed numbers into the thousands using these 7 symbols: I for 1, V for 5, X for 10, L for 50, C for 100, D for 500 and M for 1000.
5079#5164 SBLNGS CHLDRN 5164

○○○ root/log ⪢⪢
Root and log are the inverse operations of exponentiation. b˄p = y ┆ y˅p = b ┆ y⍻b = p
5079#5118 SBLNGS CHLDRN 5118

○○○ Sandwich rule ⪢⪢
Let f, g and h be three functions satisfying f(x)≤g(x)≤h(x) in some delected neighborhood of x=a. If ⎾lim f(x)⏐x￫a⏌=⎾lim h(x)⏐x￫a⏌= L then ⎾lim g(x)⏐x￫a⏌= L.
5079#4821 SBLNGS CHLDRN 4821

Proverbs and Quotes in English
Short sentences and easy words.

○○○ Sandwich rule ⪢⪢
Let f, g and h be functions such that f(x)≤g(x)≤h(x) for all x in some neighborhood of c and such that f(c)=g(c)=h(c). If f and h are conyinuous at c, then so is g.
5079#6899 SBLNGS CHLDRN 6899

○○○ Sum/product/quotient rule ⪢⪢
Suppose that ⎾lim f(x)⏐x￫a⏌= L and ⎾lim g(x)⏐x￫a⏌= M. Then the following rules apply. ❶ Sum Rule ⎾lim f(x)+g(x)⏐x￫a⏌= L+M ❷ Product Rule ⎾lim f(x)·g(x)⏐x￫a⏌= L·M ❸ Quotient Rule ⎾lim f(x)/g(x)⏐x￫a⏌= L/M, provided that M≠0.
5079#4820 SBLNGS CHLDRN 4820

○○○ Sum/product/reciprocal rules ⪢⪢
Sum rule. If f and g are continuous at c then f+g is continuous at c.┆Product rule. If f and g are continuous at c then f·g is continuous at c.┆Reciprocal rule. If f is continuous at c and f(c) is not 0 then 1/f is continuous at c.
5079#5331 SBLNGS CHLDRN 5331

○○○ symmetry ⪢⪢
_____ is when an object can be folded in half and both of the sides are the same.
5079#28249 SBLNGS CHLDRN 28249

• Taylor and Maclaurin Series
5079#3574 SBLNGS CHLDRN 3574

○○○ topology ⪢⪢
5079#3793 SBLNGS CHLDRN 3793

◌◌◌ triangle
5079#3961 SBLNGS CHLDRN 3961

○○○ trigonometry ⪢⪢
5079#32670 SBLNGS CHLDRN 32670

○○○ β ⪢⪢
5079#20855 SBLNGS CHLDRN 20855

○○○ π ⪢⪢
5079#20671 SBLNGS CHLDRN 20671

○○○ Φ ≈ 1.618 ⪢⪢
Keplar observed that the relationship between a number in the Fibonacci sequence and the previous number more and more closely approaches the irrational number Φ, the longer the sequence is continued. #math #bookmark
5079#9464 SBLNGS CHLDRN 9464

○○○ ψ ≈ 137.5° ⪢⪢
5079#9465 SBLNGS CHLDRN 9465

○○○ ψ ≈ 137.5° ⪢⪢
As angles smaller than 180° proved to be more handy in practice, the smaller of the resultant angles is usually called golden angle. #math
5079#28243 SBLNGS CHLDRN 28243