- •Arithmetic
- •How the use of numbers began
- •Exercises
- •How we read and write numbers
- •Exercises
- •Adding, subtracting, multiplying and dividing the whole numbers
- •Exercises
- •Fractions and their meaning
- •Exercises
- •Types of fractions
- •Exercises
- •Addition, subtraction, multiplication and division of fractions
- •Exercises
- •Changing fractions
- •Decimal fractions
- •Exercises
- •Adding, subtracting, multiplying and dividing decimal fractions
- •Exercises
- •What is per cent?
- •Exercises
- •Scale drawing
- •Exercises
- •Exercises
- •Algebra
- •The nature of algebra
- •Exercises
- •Signs used in algebra
- •Exercises
- •Equations
- •Exercises
- •Monomial and polynomial
- •Exercises
- •Factors, coefficients and combining terms
- •Exercises
- •The formula
- •Exercises
- •Systems of two linear equations1 in two unknowns
- •Exercises
- •Squares and square roots
- •Exercises
- •Logarithms
- •Exercises
- •The slide-rule
- •Exercises
- •Geometry
- •Points and lines
- •Measuring and constructing angles with a protractor
- •Exercises
- •Kinds of polygons
- •Exercises
- •Circles
- •Exercises
- •Geometric solids
- •Exercises
- •Symmetry
- •Exercises
- •Similar fioures
- •Exercises
- •Trigonometry
- •Trigonometry and its application
- •Exercises
- •Trigonometric functions
- •Exercises
- •Measurement of angles
- •Exercises
- •Functions of complementary angles
- •Exercises
- •The solution of right triangles
- •Exercises
- •Tables of values of the trigonometric functions
- •Exercises
- •Exercises
- •Supplementary reading
- •Pythagoras
- •Leibnitz
- •Sophia kovalevskaya
- •Nikolai lobachevsky
- •Mathematician No. 1
- •About common fractions
- •Mathematics—handyman for all sciences
- •Ordinary vs. Binary numbers
- •Appendix signs used in mathematics
- •Short mathematics dictionary
- •English – russian vocabulary of mathematical terms
Squares and square roots
To square a number1, you have learned, you must multiply that number by itself. The square root of a number is just the opposite. When you find the square root of a number, you are finding what number multiplied by itself gives you the number you began with2. The sign for the square root is √. Thus, the square root of 25 is represented by √25. 25 is a perfect square. That is, a whole number (5) multiplied by itself will give you 25. Most numbers are not perfect
squares. In that case, to get the square root of a number we may either find it by taking an arithmetic square root or by using a table.
The process of finding a root is known as evolution; it is the inverse of involution, because by the aid of this process we try to find that which is given only when raising a number to a power (viz. the base of the power), while the data given is just what is sought for3 raising a number to a power (viz. the power itself). Therefore the accuracy of the root taken may always be checked by raising the number to the power4. For instance, in order to check the equality: 3√125=5, it is sufficient to cube 5; obtaining the quantity under the radical
sign, we conclude that the cube root of 125 has been found correctly.
Notes:
1 to square a number — чтобы возвести число в квадрат
2 the number you began with — зд. исходное число
3 what is sought for — искомое
4 by raising the number to the power — возведением числа в степень
Exercises
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Read the following words paying attention to the pronunciation:
inverse, learn, perfect, order, for, opposite, not, must, number, thus.
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Form Participles using the following verbs:
to square, to use, to raise, to multiply, to find, to check, to give, to begin, to obtain, to get, to take, to be.
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Make up sentences of your own using the words and expressions given below:
to raise to power, to obtain the quantity, to square the number, to take an arithmetic square root, to use a table may be checked, conclude.
IV. Answer the following questions:
1. What operation should be performed to square a number? 2. What is a perfect square? 3. What do we do to get the square root of a number? 4. What is the process of finding a root called? 5. How do we check the accuracy of a root?
V. Translate into Russian:
Tables of squares are used by architects and engineers in working with squares of number. If you have a table of squares, you can find the approximate square root of any number. Sometimes it is not easy to find a square root by
inspection. If a table of squares is not at hand another method may be used.
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Translate into English:
Чтобы возвести в квадрат число, надо умножить это число на самое себя. Извлечение квадратного корня — это действие обратное возведению в квадрат. Чтобы получить квадратный корень числа, мы можем пользоваться специальной таблицей. Правильность извлечения квадратного корня можно проверить, возведя в квадрат подкоренное
выражение; если получится данное число, то корень найден правильно.
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