# Calculations 計算

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## spoken calculations

2+2=4

Two and two is four.  (informal)
Two plus two equals/is four.  (formal)

7-4=3

Four from seven is/leaves three.  (informal)
Seven take away four is/leaves three.  (informal)
Seven minus four equals/is three.  (formal)

3×4=12

Three fours are twelve.  (informal)
Three times four is twelve.  (informal)
Three multiplied by four equals/is twelve.  (formal)

to calculate on paper

 146 ×281 29200 11680 146 41026

setting:
A hundred and forty-six times two hundred and eighty-one.

beginning:
Put down two noughts/zeros. Tow sixes are twelve; put down two and carry one; two fours are eight and one are nine; two ones are two.

next line:
Put down one nought/zero. eight sixes are forty-eight; put down eight and carry four; eight fours are thirty-two and four is thirty-six; put down six and carry three; eight ones are eight and three is eleven.

next line:
One times 146 is 146.

Six and nought/zero and nought/zero is six; eight and four and nought/zero is twelve; put down two and carry one; six and two are eight and one is nine and one is ten; put down nought/zero carry one; nine and one are ten and one is eleven; put down one and carry one; two and one are three and one are four.

total:
Forty-one thousand and twenty-six.

9÷3=3

Three(s) into nine goes three (times).  (informal, especially BrE)
Nine divided by three equals/is three. (formal)

$$\frac{2}{7} \div \frac{2}{3} = \frac{3}{7}$$

Two sevenths divided by two thirds equals/is three sevenths.

10 ÷ 3 = 3 r 1

Oh, there’s a remainder!

Ten divided by three equals/is three with a remainder of one.

## power and root  累乗と根

$$x^n$$

x to the nth power

$$x^{-n}$$

x to the minus nth power

$$5^2$$

five squared
five to the second power

$$7^3$$

seven cubed
seven to the third power

$$a^2 + b^3 = c^4$$

A squared plus b cubed equals c to the fourth power.

$$\sqrt[n]{x}$$

the nth root of x

$$\sqrt{7}$$

the square root of seven
the second root of seven
root seven

$$\sqrt[3]{25}$$

the cube root of twenty-five
the third root of twenty-five

$$\sqrt[5]{432}$$

the fifth root of four hundred thirty-two

## Multiplication / Times tables　九九

Please tell me the four times table.

Let’s begin our multiplication table of four.

The four times table starts; “Four times zero is zero, four times one is one, …”

### 1 Times Table

 0 X 1 = 0 Zero times one is zero. 1 X 1 = 1 One times one is one. 2 X 1 = 2 Two times one is two. 3 X 1 = 3 Three times one is three. 4 X 1 = 4 Four times one is four. 5 X 1 = 5 Five times one is five. 6 X 1 = 6 Six times one is six. 7 X 1 = 7 Seven times one is seven. 8 X 1 = 8 Eight times one is eight. 9 X 1 = 9 Nine times one is nine. 10 X 1 = 10 Ten times one is ten. 11 X 1 = 11 Eleven times one is eleven. 12 X 1 = 12 Twelve times one is twelve.

### 2 Times Table

 0 X 2 = 2 Zero times two is zero. 1 X 2 = 2 One times two is two. 2 X 2 = 4 Two times two is four. 3 X 2 = 6 Three times two is six. 4 X 2 = 8 Four times two is eight. 5 X 2 = 10 Five times two is ten. 6 X 2 = 12 Six times two is twelve. 7 X 2 = 14 Seven times two is forteen. 8 X 2 = 16 Eight times two is sixteen. 9 X 2 = 18 Nine times two is eighteen. 10 X 2 = 20 Ten times two is twenty. 11 X 2 = 22 Eleven times two is twenty-two. 12 X 2 = 24 Twelve times two is twenty-four.

### 3 Times Table

 0 X 3 = 0 Zero times three is zero. 1 X 3 = 3 One times three is three. 2 X 3 = 6 Two times three is six. 3 X 3 = 9 Three times three is nine. 4 X 3 = 12 Four times three is twelve. 5 X 3 = 15 Five times three is fifteen. 6 X 3 = 18 Six times three is eighteen. 7 X 3 = 21 Seven times three is twenty-one. 8 X 3 = 24 Eight times three is twenty-four. 9 X 3 = 27 Nine times three is twenty-seven. 10 X 3 = 30 Ten times three is thirty. 11 X 3 = 33 Eleven times three is thirty-three. 12 X 3 = 36 Twelve times three is thirty-six.

### 4 Times Table

 0 X 4 = 0 Zero times four is zero. 1 X 4 = 4 One times four is four. 2 X 4 = 8 Two times four is eight. 3 X 4 = 12 Three times four is twelve. 4 X 4 = 16 Four times four is sixteen. 5 X 4 = 20 Five times four is twenty. 6 X 4 = 24 Six times four is twenty-four. 7 X 4 = 28 Seven times four is twenty-eight. 8 X 4 = 32 Eight times four is thirty-two. 9 X 4 = 36 Nine times four is thirty-six. 10 X 4 = 40 Ten times four is forty. 11 X 4 = 44 Eleven times four is forty-four. 12 X 4 = 48 Twelve times four is forty-eight.

### 5 Times Table

 0 X 5 = 0 Zero times five is zero. 1 X 4 = 5 One times five is five. 2 X 5 = 10 Two times five is ten. 3 X 5 = 15 Three times five is fifteen. 4 X 5 = 20 Four times five is twenty. 5 X 5 = 25 Five times five is twenty-five. 6 X 5 = 30 Six times five is thirty. 7 X 5 = 35 Seven times five is thirty-five. 8 X 5 = 40 Eight times five is forty. 9 X 5 = 45 Nine times five is forty-five. 10 X 5 = 50 Ten times five is fifty. 11 X 5 = 55 Eleven times five is fifty-five. 12 X 5 = 60 Twelve times five is sixty.

### 6 Times Table

 0 X 6 = 0 Zero times six is zero. 1 X 6 = 6 One times six is six. 2 X 6 = 12 Two times six is twelve. 3 X 6 = 18 Three times six is eighteen. 4 X 6 = 24 Four times six is twenty-four. 5 X 6 = 30 Five times six is thirty. 6 X 6 = 36 Six times six is thirty-six. 7 X 6 = 42 Seven times six is forty-two. 8 X 6 = 48 Eight times six is forty-eight. 9 X 6 = 54 Nine times six is fifty-four. 10 X 6 = 60 Ten times six is sixty. 11 X 6 = 66 Eleven times six is sixty-six. 12 X 6 = 72 Twelve times six is seventy-two.

### 7 Times Table

 0 X 7 = 0 Zero times seven is zero. 1 X 7 = 7 One times seven is seven. 2 X 7 = 14 Two times seven is forteen. 3 X 7 = 21 Three times seven is twenty-one. 4 X 7 = 28 Four times seven is twenty-eight. 5 X 7 = 35 Five times seven is thirty-five. 6 X 7 = 42 Six times seven is forty-two. 7 X 7 = 49 Seven times seven is forty-nine. 8 X 7 = 56 Eight times seven is fifty-six. 9 X 7 = 63 Nine times seven is sixty-three. 10 X 7 = 70 Ten times seven is seventy. 11 X 7 = 77 Eleven times seven is seventy-seven. 12 X 7 = 84 Twelve times seven is eighty-four.

### 8 Times Table

 0 X 8 = 0 Zero times eight is zero. 1 X 8 = 8 One times eight is eight. 2 X 8 = 16 Two times eight is sixteen. 3 X 8 = 24 Three times eight is twenty-four. 4 X 8 = 32 Four times eight is thirty-two. 5 X 8 = 40 Five times eight is forty. 6 X 8 = 48 Six times eight is forty-eight. 7 X 8 = 56 Seven times eight is fifty-six. 8 X 8 = 64 Eight times eight is sixty-four. 9 X 8 = 72 Nine times eight is seventy-two. 10 X 8 = 80 Ten times eight is eighty. 11 X 8 = 88 Eleven times eight is eighty-eight. 12 X 8 = 96 Twelve times eight is ninety-six.

### 9 Times Table

 0 X 9 = 0 Zero times nine is zero. 1 X 9 = 9 One times nine is nine. 2 X 9 = 18 Two times nine is eighteen. 3 X 9 = 27 Three times nine is twenty-seven. 4 X 9 = 36 Four times nine is thirty-six. 5 X 9 = 45 Five times nine is forty-five. 6 X 9 = 54 Six times nine is fifty-four. 7 X 9 = 63 Seven times nine is sixty-three. 8 X 9 = 72 Eight times nine is seventy-two. 9 X 9 = 81 Nine times nine is eighty-one. 10 X 9 = 90 Ten times nine is ninety. 11 X 9 = 99 Eleven times nine is ninety-nine. 12 X 9 = 108 Twelve times nine is one hundred oh eight.

### 10 Times Table

 0 X 10 = 0 Zero times ten is zero. 1 X 10 = 10 One times ten is ten. 2 X 10 = 20 Two times ten is twenty. 3 X 10 = 30 Three times ten is thirty. 4 X 10 = 40 Four times ten is forty. 5 X 10 = 50 Five times ten is fifty. 6 X 10 = 60 Six times ten is sixty. 7 X 10 = 70 Seven times ten is seventy. 8 X 10 = 80 Eight times ten is eighty. 9 X 10 = 90 Nine times ten is ninety. 10 X 10 = 100 Ten times ten is one hundred. 11 X 10 = 110 Eleven times ten is one hundred and ten. 12 X 10= 120 Twelve times ten is one hundred and twenty.

### 11 Times Table

 0 X 11 = 0 Zero times eleven is zero. 1 X 11 = 11 One times eleven is eleven. 2 X 11 = 22 Two times eleven is twenty-two. 3 X 11 = 33 Three times eleven is thirty-three. 4 X 11 = 44 Four times eleven is forty-four. 5 X 11 = 55 Five times eleven is fifty-five. 6 X 11 = 66 Six times eleven is sixty-six. 7 X 11 = 77 Seven times eleven is seventy-seven. 8 X 11 = 88 Eight times eleven is eighty-eight. 9 X 11 = 99 Nine times eleven is ninety-nine. 10 X 11 = 110 Ten times eleven is one hundred and ten. 11 X 11 = 121 Eleven times eleven is one hundred and twenty-one. 12 X 11= 132 Twelve times eleven is one hundred and thirty-two.

### 12 Times Table

 0 X 12 = 0 Zero times twelve is zero. 1 X 12 = 12 One times twelve is twelve. 2 X 12 = 24 Two times twelve is twenty-four. 3 X 12 = 36 Three times twelve is thirty-six. 4 X 12 = 48 Four times twelve is forty-eight. 5 X 12 = 60 Five times twelve is sixty. 6 X 12 = 72 Six times twelve is seventy-two. 7 X 12 = 84 Seven times twelve is eighty-four. 8 X 12 = 96 Eight times twelve is ninety-six. 9 X 12 = 108 Nine times twelve is one hundred oh eight. 10 X 12 = 120 Ten times twelve is one hundred and twenty. 11 X 12 = 132 Eleven times twelve is one hundred and thirty-two. 12 X 12= 144 Twelve times twelve is one hundred and forty-four.

## Find the area of this figure. この図形の面積を求めよ

### πr2  円の面積

The area of a circle is equal to pi times the radius squared.

### b × h  平行四辺形の面積

The area of a parallelogram is equal to its base (b) times the height (h).

### (b × h)÷2  三角形の面積

The area of a triangle is equal to its base (b) times the height (h) divided by two.

### {(a+b)×h}÷2  台形の面積

The area of a trapezoid is equal to the sum of both bases (a+b) times the height (h) divided by two.

## Find the volume of this box. この箱の体積を求めよ

### l3  立方体の体積

The volume of a cube is equal to the length of one of its sides cubed.

### (b×l×h)÷3  四角錐の体積

The volume of a pyramid is equal to the area of its base (b times l) times the height (h) divided by three.

### (4×π×r3)÷3  球の体積

The volume of a sphere is equal to four times pi times the radius cubed divided by three.

## mathematical expressions  数式

4x + 2x = 6x

Four x plus two x equals six x.

y = – 6x2 + 8x + 3

Y equals  negative/minus six x squared plus eight x plus three.

(a + b)2 = a2 + 2ab + b2

[The quantity] A plus b squared equals a squared plus two (times) a (times) b plus b squared.

(a+b)(a-b)=a2 – b2

[The quantity] A plus b times [the quantity] a minus b equals a squared minus b squared.

y = (4x2 – 12x + 9)÷(2x – 3)

Y equals [the quantity] four x squared minus twelve x plus 9 divided by [the quantity] two x minus three.

a2 + b2 ≧ 2(a – b – 1)

A squared plus b squared is greater than or equal to two times [the quantity] a minus b minus one.

$$\sqrt{4} + \sqrt{16} = 6$$

The square root of four plus the square root of sixteen equals six.

$$a^2 + b^3 = \sqrt[3]{c}$$

A squared plus b cubed equals the cube root of c.

$$y = f(x)$$

Y is[equals] a function of x.

3 : 7 = 6 : 14

Three is to seven as six is to fourteen.

$$\sum_{n=0}^\infty a_n$$

Summation from zero to infinity of a sub n.

シグマのn=0から無限のエーエヌ

$$\int_a^b f(x) dx$$

Integral of a function of x with respect to x from x equals a to x equals b.

インテグラルa, bエフエックスディエックス

## relational operator 関係演算子

A ≠ B

A is not equal to B.
A does not equal B.
AはBと等しくない

A > B

A is greater than B.
AはBより大きい

A < B

A is less than B.
AはBより小さい

A ≧ B

A is greater than or equal to B.
AはBより大きいか等しい

A ≦ B

A is less than or equal to B.
AはBより小さいか等しい

\begin{eqnarray}
30未満 = \begin{cases}
under\,\,30 \\
below\,\,30 \\
less\,\,than\,\,30 \\
\end{cases}
\end{eqnarray}
\begin{eqnarray}
30以下 = \begin{cases}
not\,\,more\,\,than\,\,30 \\
30\,\,and\,\,(or)\,\,less \\
30\,\,and\,\,(or)\,\,under \\
30\,\,and\,\,(or)\,\,below \\
\end{cases}
\end{eqnarray}
\begin{eqnarray}
30より多く = \begin{cases}
more\,\,than\,\,30 \\
over\,\,30 \\
above\,\,30 \\
\end{cases}
\end{eqnarray}
\begin{eqnarray}
30以上 = \begin{cases}
not\,\,less\,\,than\,\,30 \\
30\,\,and\,\,(or)\,\,more \\
30\,\,and\,\,(or)\,\,above \\
30\,\,and\,\,(or)\,\,over \\
\end{cases}
\end{eqnarray}

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