Mathematics

• Reading Numbers
• Reading Equations
• Types of Numbers
• Geometry
  • Shapes
  • Relationships
• Symbols
• Functions
  • Roots
  • Exponents and Logarithms
  • Fractions
  • Trigonometric Functions
  • Sums
  • Combinations
• Calculus
• Terminology
  • Definitions
  • Theorems
  • Corollaries
• Graphs

• Easy to confuse
  • 13 : thirteen and 30 : thirty
  • 14 : fourteen and 40 : forty
  • 15 : fifteen and 50 : fifty
  • 16 : sixteen and 60 : sixty
  • 17 : seventeen and 70 : seventy
  • 18 : eighteen and 80 : eighty
  • 19 : nineteen and 90 : ninety
• Unlike Japanese, numbers in English are grouped in threes, i.e.,
  • 1,000 : one thousand
  • 10,000 : ten thousand
  • 100,000 : one hundred thousand
  • 1,000,000 : one million
  • 1,000,000,000 : one billion
  • 1,000,000,000,000 : one trillion

• Examples:
  • 23,845 : twenty three thousand, eight hundred and forty five
  • 943,548 : nine hundred forty three thousand, five hundred and forty eight
  • 157,000,000 : one hundred and fifty seven million
  • 200,010 : two hundred thousand and ten

• Special cases
  • There are two ways to read most 4-digit numbers
  • 3488 : three thousand four hundred and eighty eight OR thirty four hundred and eighty eight
  • 1800 : one thousand eight hundred OR eighteen hundred
  • 2000 : two thousand NOT twenty hundred

• Decimal Numbers
  • 3.4 : three point four
  • 9.394 : nine point three nine four

• Fractions
  • Fractions are read from top to bottom in English, e.g.,
  • 1/2 : one half
  • 2/3 : two thirds
  • 3/4 : three fourths
  • 4/5 : four fifths
  • 1/6 : one sixth
  • 3/7 : three sevenths
  • 1/8 : one eighth
  • 1/9 : one ninth
  • 7/10 : seven tenths
  • 3/31 : three thirty firsts
  • 5/42 : five forty seconds
  • 9/100 : nine hundredths
  • 77/1000 : seventy seven thousandths
  • 8/1000000 : eight millionths
  • 230/39001 : two hundred and thirty over thirty nine thousand and one

• Scientific Notation
  • 1.4 x 10² : one point four times ten to the second
  • 5.89 x 10³ : five point eight nine times ten to the third
  • 4.3 x 10^-1 : four point three times ten to the minus one
  • 3.9 x 10^-3 : three point nine times ten to the minus three

• 5 × 3 = 15 : five times 3 equals 15
• ab = c : the product of a and b is equal to c
• 9 + 4 > 4 : nine plus four is greater than 4
• a ≧ 3 - 9 : a is greater than or equal to 3 minus nine
• 8 ÷ 3 < 12 : eight divided by 3 is less than twelve
• b ≦ 8.4 : b is less than or equal to 8 point 4

• negative number (＜0)
• non-positive number (≦0)
• positive number (＞0)
• non-negative number (≧0)
• integer (整数)
• real number (実数)
• complex number (複素数)
• rational number (有理数)
• irrational number (無理数)

Two-dimensional shapes

• circle : 円
• diameter : 直径
• radius : 半径
• center : 中心
• tangent line : 接線
• square : 正方形
• rectangle : 長方形
• triangle : 三角形
  • equilateral triangle : 正三角形
  • right triangle : 直角三角形
  • isosceles triangle : 二等辺三角形
• angle : 角度
• vertex (two vertices) : 頂点
• trapezoid : 台形
• side : 辺
• polygon : 多角形
  • pentagon : 五角形
  • hexagon : 六角形
  • heptagon : 七角形
  • octagon : 八角形
  • nonagon : 九角形
  • decagon : 十角形
• hyperbola : 双曲線
• arc : 弧
• segment : 線分
• parabola : 放物線
• ellipse : 楕円
• focus (two foci) : 焦点

Three-dimensional shapes

• cube : 立方体
• rectangular prism : 直方体
• cylinder : 柱体
• cone : 円錐
• sphere : 球
• polyhedron : 多面体
  • tetrahedron : 四面体
  • pentahedron : 五面体
  • hexahedron : 六面体
  • octahedron : 八面体
  • decahedron : 十面体
• pyramid : ピラミッド

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Relationships

• perpendicular
  
  
  
  
  • Line A is perpendicular to line B.
  • Lines A and B are perpendicular.
  • Line A is a vertical line.
  • Line B is a horizontal line.
  
  
• parallel
  
  
  
  
  
  • Side AB is parallel to side DC.
  
  
• tangent
  
  
  • Line AB is tangent to the ellipse at point C.
  • The two circles are tangent.
  
  
  
• intersect
  
  
  • The parabola intersects triangle ABC at points D and E.
    
    
  
  
• bisect
  
  
  
  
  • Segment AB bisects side CD.
  • Angle ABC is bisected by segment BD.
  
  
• circumscribe
  
  
  
  
  • The circle is circumscribed around the triangle.
  • The circumscribed circle has a diameter of 4 cm.
  
  
• inscribe
  
  
  
  
  • The circle is inscribed inside the square.
  • The inscribed circle has a radius of 2 cm.

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Symbols

• ( ) : parenthesis, parentheses
• [ ] : bracket, brackets
• { }: brace, braces
• * : asterisk
• & : ampersand
• ~ : tilde
• / : slash
• ＼ : back slash
• ^ : circumflex
• # : number sign
• @ : at
• ' : apostrophe
• ` : grave
• _ : underscore
• | : bar

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Functions

Roots

• : the square root of 2, root 2
• : the cube root of x
• : the fourth root of y

Exponents and Logarithms

• x² : x squared
• x³ : x cubed
• y⁴ : y to the fourth power
• y⁵ : y to the fifth power
• y⁶ : y to the sixth [power]
• x⁷ : x to the seventh [power]
• x⁸ : x to the eighth [power]
• y⁹ : y to the ninth [power]
• y¹⁰ : y to the tenth [power]
• y³⁹ : y to the thirty-ninth [power]
• z^-1 : z to the minus one
• z^-12 : z to the minus twelve
• e^-x : e to the minus x
• 2^x : 2 to the power of x, 2 to the x
• log (z) : log z
• log₂ (c) : log to the base 2 of c

Fractions

• 1/x : one over x, the reciprocal of x
• y/z : y over z
• (x+y)/z : x plus y all over z

Trigonometric Functions

• sin (x) : sine x
• cos (x) : cosine x
• tan (x) : tangent x
• csc (x) : cosecant x
• sec (x) : secant x
• cot (x) : cotangent x

Sums

• : the sum from n equals 0 to 10 of n cubed

Combinations

• sin²(x/y) : sine squared x over y
• : one over the square root of x plus 3
• f(x) = sin²(x) + e^-y : f of x is equal to sine squared x plus e to the minus y
• 10^x/y : ten to the power of x over y

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Calculus

• : the integral from 0 to t of x cubed d x
• : the derivative of f with respect to t, d f by d t
• : the limit as x goes to infinity of x over x plus one

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Terminology （用語）

Definitions

• definition （定義）
• define （定義する）
• denote （意味する）
• x is defined to be the sum of a and b.
• The definition of x is the sum of a and b.
• The height of the triangle is denoted by h.
• Definition: If a₀, a₁, . . ., a_n are real numbers, then the function f given by
  f(x) = a₀ xⁿ + a₁ x^n-1 + . . . + a_n,
  where n is a positive integer or zero, is called a polynomial. If a₀ ≠ 0 then the polynomial is of degree n.
  • . . . : dot dot dot
  • a₀ ≠ 0 : a sub zero is not equal to zero
  • polynomial （多項式）
  • function （関数）

Theorems

• theorem （定理）
• proof （証明）
• algebra （代数学）
• The fundamental theorem of algebra.
  If a₀, . . .,a_n are any real or complex numbers (a₀ ≠ 0), then the equation
  a₀ xⁿ + a₁ x^n-1 + . . . + a_n = 0
  has at least one root, real or complex.

Corollaries

• corollary （系）
• Corollary to the fundamental theorem of algebra There exist numbers (real or complex) r₁, r₂, . . .,r_n (not necessarily distinct) such that f(x) factors uniquely into linear factors:
  f(x) = a₀(x-r₁)(x-r₂) . . . (x-r_n).
• exist （存在する）
• distinct （異なる）
• linear （線形）
• factor （因数、因数に分解する）
• uniquely （唯一）

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Graphs

• x-axis
• abscissa
• y-axis
• ordinate
• the curve has a local maximum at x=8
• the coordinates of the global minimum are x=8 and y=5
• the slope of the line is 7
• the line's y-intercept is -4

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© 2003 David Asano. All Rights Reserved