Arithmetic Operations with Natural Numbers

Whole (natural) numbers

• Natural numbers are those used when counting single items.
• Natural numbers form a natural series that goes on endlessly: 1, 2, 3, 4, 5...

Arithmetical operations

• Addition is finding a sum of numbers.

• In the equation 11 + 6 = 17, 11 and 6 are addends and 17 is the sum.

• Changing the order of addends doesn't change the sum.

• Subtraction finds an addend using the sum and another addend.

• In the equation 17 - 6 = 11, 17 is the minuend, 6 is the subtrahend, and 11 is the difference.

• Multiplication repeats a multiplicand a number of times indicated by a multiplier.

• The result of multiplication is called a product.

• In the equation 12 x 4 = 12 + 12 + 12 + 12 = 48, 12 is the multiplicand, 4 is the multiplier, and 48 is the product.

• Changing the order of the multiplicand and multiplier doesn't change the product.

• Multiplicand and multiplier are usually called factors or multipliers.

• Division finds a factor using the product and another factor.

• In the equation 48 : 4 = 12, 48 is the dividend, 4 is the divisor, and 12 is the quotient.

• When dividing integers, the quotient isn't always a whole number, it can be a fraction.

• If the quotient is a whole number, the numbers are divisible without a remainder.

• A division with remainder example: 23 = 5 * 4 + 3, where 3 is the remainder.

• Raising to a power repeats a number as a factor a certain number of times.

• The number being repeated is the base, the number of factors is the exponent, and the result is the value of the power.

• Exponentiation is written as repeated multiplication.

• 3 to the power of 5 or 3^5 is 3 * 3 * 3 * 3 * 3 = 243.

• In the equation 3^5 = 243, 3 is the base, 5 is the exponent, and 243 is the value.

• The second power is called a square, the third a cube, and the first power of any number is the number itself.

• Extraction of a root finds the base of a power given the power and its exponent.

• In the equation 5√243 = 3, 243 is the radicand, 5 is the index or degree of the root, and 3 is the value of the root.

• The second root is called a square root, while the third root is a cube root.

• Addition/subtraction and multiplication/division, raising to a power/extraction of a root are two by two mutually inverse operations.

Order of operations and use of brackets

• The order of operations is:
  • Raising to a power and extraction of a root (one after another).
  • Multiplication and division (left to right).
  • Addition and subtraction (left to right).
• Operations inside brackets are executed first, following the same order, then the rest of the operations outside.
• Example expression: (10 + 2 * 3^3) + 4^3 - (16 : 2 – 1) * 5 – 150 : 5^2 = 57

Laws of addition and multiplication

• Commutative law of addition: m + n = n + m (order of addends doesn't change the sum).
• Commutative law of multiplication: m * n = n * m (order of factors doesn't change the product).
• Associative law of addition: (m + n) + k = m + (n + k) = m + n + k (grouping of addends doesn't affect the sum).
• Associative law of multiplication: (m * n) * k = m * (n * k) = m * n * k (grouping of factors doesn't affect the product).
• Distributive law of multiplication over addition: (m + n) * k = m * k + n * k.
• The distributive law expands the rules of operations with brackets.

Prime and composite numbers

• Prime numbers: Numbers divisible only by 1 and themselves.
• Composite numbers: Numbers with factors other than 1 and themselves.
• There is an infinite amount of prime numbers, some examples include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

Factorization

• Factorization is the decomposition of a composite number into its prime factors.
• Any composite number can be presented as a product of prime factors in a unique way.
• Examples, 48 = 2 * 2 * 2 * 2 * 3, 225 = 3 * 3 * 5 * 5, 1050 = 2 * 3 * 5 * 5 * 7
• For large numbers, use a systematic approach to find prime factors.

Greatest Common Factor (GCF)

• A common factor of some numbers is a number which is a factor of each of them.
• The greatest common factor (GCF) is the largest factor shared by those numbers.
• The find the GCF:
  • Express the numbers as a product of prime factors.
  • Write the prime factors with powers.
  • Write out all common factors in these factorizations.
  • Take the smallest power of each of the factored numbers.
  • Multiply these powers together.
• To find the GCF for 168, 180, and 3024:
  • 168 = 2^3 * 3^1 * 7^1
  • 180 = 2^2 * 3^2 * 5^1
  • 3024 = 2^4 * 3^3 * 7^1
  • GCF = 2^2 * 3^1 = 12

Least Common Multiple (LCM)

• A common multiple of given numbers is a number that is divisible by each of the given numbers.
• The least common multiple (LCM) is the smallest of these multiples.
• To find the LCM, complete the following steps:
  • Express each number as a product of its prime factors.
  • Write powers of all prime factors in the factorization.
  • Write out all prime factors, presented in at least one of the numbers.
  • Take the greatest power of each factored number
  • Multiply these powers.
• Example: Find LCM for numbers 168, 180, and 3024
  • 168 = 2^3 * 3^1 * 7^1
  • 180 = 2^2 * 3^2 * 5^1
  • 3024 = 2^4 * 3^3 * 7^1
  • LCM = 2^4 * 3^3 * 5^1 * 7^1 = 15120

Divisibility criteria

• Divisibility by 2: Last digit is 0 or divisible by 2.
  • Numbers divisible by 2 are even, otherwise they are odd.
• Divisibility by 4: Last two digits are zeros or form a number divisible by 4.
• Divisibility by 8: Last three digits are zeros or form a number divisible by 8.
• Divisibility by 3 and 9: Sum of digits is divisible by 3 or 9, respectively.
• Divisibility by 6: Divisible by both 2 and 3.
• Divisibility by 5: Last digit is 0 or 5.
• Divisibility by 25: Last two digits are zeros or form a number divisible by 25.
• Divisibility by 10: Last digit is 0.
• Divisibility by 100: Last two digits are zeros.
• Divisibility by 1000: Last three digits are zeros.
• Divisibility by 11: The sum of digits in even places is equal to or differs by a multiple of 11 from the sum of digits in odd places.

Simple fractions

• A simple fraction represents a part of a unit, expressed as a numerator and a denominator.
• The denominator indicates the number of equal parts into which the unit is divided.
• The numerator indicates the number of these parts taken.
• If the numerator is less than the denominator, the fraction is less than 1 and is called a proper fraction.
• If the numerator is equal to the denominator, the fraction is equal to 1.
• If the numerator is greater than the denominator, the fraction is greater than 1 and is called an improper fraction.
• If the numerator is divisible by the denominator, the fraction equals a quotient.
• If division results in a remainder, the improper fraction is a mixed number.
• For example, convert 8 5/9 into a vulgar fraction.
  • 8 * 9 = 72
  • 72 + 5 = 77
  • So 8 5/9 = 77/9
• Reciprocal fractions are two fractions whose product is 1, e.g. 3/7 and 7/3.

Operations with simple fractions

• Extension of a fraction: Multiplying the numerator and denominator by the same non-zero number doesn't change the fraction's value.
• E.g. 5/9 = (5 * 7) / (9 * 7) = 35/63
• Cancellation of a fraction: Dividing the numerator and denominator by the same non-zero number doesn't change the fraction's value.
  • This is also known as reducing to the lowest term.
  • E.g. 18/27 = (2 * 9) / (3 * 9) = 2/3
• To compare fractions with the same numerators, the one with the smaller denominator is greater.
• To compare fractions with the same denominators, the one with the larger numerator is greater.
• If both numerators and denominators are different, extend them to have the same denominators.
• When comparing 2/3 and 7/10 = (2 * 10) / (3 * 10) and (7 * 3) / (10 * 3) = 20/30 and 21/30 and we see that 2/3 < 7/10 because 20/30 < 21/30
• The process of making the denominators the same is called reducing the fractions to a common denominator.
• When adding (or subtracting) fractions with the same denominators, you can add (or subtract) the numerators and keep the same denominator.
• With fractions of different denominators: Reduce to a common denominator, then add them.
• At addition of mixed numbers a sum of integer parts and a sum of fractional parts are found separately.
• Subtracting mixed numbers: convert to improper fractions, subtract them, and convert the result back into a mixed number.
• Multiplying fractions: Multiply their numerators and denominators separately and divide the first product by the second.
• Dividing fractions: Multiply by a reciprocal fraction.

Decimal fractions (decimals)

• Decimal fractions are the fractions with denominators of 10, 100, 1000, etc
• Calculations are performed, based on the same system used for integers.
• Digits after the decimal point are called decimal places.
• Decimals can be easily converted into vulgar fractions. The digits after the decimal point is a numerator, and the n-th power of 10 (n - a quantity of decimal places. - is a denominator.

Rules for fractions

• Adding zeroes to the right of a decimal fraction doesn't change its value, 13.6 = 13.6000.
• Rejecting zeroes located at the end of a decimal fraction doesn't change its value.
• To increase a decimal fraction by 10, 100, 1000, times, move the decimal point one, two, three places to the right.
• To decrease a decimal fraction by 10, 100, 1000, times, move the decimal point one, two, three places to the left.
• A repeating decimal is a decimal in which a digit or digits repeats endlessly in a pattern.
• A digit or digits is called a period of decimal and is written in brackets.
• Result of dividing 47 by 11, is 4.27272727 = 4.(27)
• When adding and subtracting decimal fractions, write the corresponding digits one under another.
• Multiplying decimal fractions: Multiply as integers, then place the decimal point in the product so that the amount of decimal places is the sum of the places in the multiplied factors.
• If decimal is less than a divisor, write zero in an integer part of a quotient and put after it a decimal point, and continue the process in the same manner.
• Dividing decimal fraction by another one: Transfer decimal points in a dividend and a divisor by the number of decimal places of divisor, i.e. make the divisor an integer, Now divide as well as in the previous case.
• To convert a decimal to a vulgar transaction: Put the number after decimal point to a numerator, and the n-th power of 10 (the quantity of decimal places) in a denominator.
• To convert a vulgar fraction to a decimal divide the numerator by the denominator.

Percents

• Percent is a hundredth part of a unit, 1% = 0.01.
• To find an indicated percent of a given number: multiply the given number by the percent and divide by 100.
• To find a number by its percent value: divide the given number by its percent value and multiply it by 100.
• To find the percent expression of one number by another: divide the first number by the second and multiply by 100.

Ratio and proportion

• Ratio is a quotient of dividing one number by another.
• Proportion is an equality of two ratios
• In the proportion a:b = c:d; a and d are border terms, b and c are middle terms.
• Main property of a proportion: product of border terms is equal to the product of its middle terms.
• With proportional values, a ratio of their values is invariable.

Integers and Rational Numbers

• Integers: Natural numbers, their negatives, and zero.
• Rational numbers: Ratios a/b of integers, where b ≠ 0.
  • Integers are rational numbers with b = 1.
  • Rational numbers are repeating decimals.
• Negative integers: Appear when subtracting the greater integer from the smaller one:
• The sign "minus" indicates that this number is negative.
• Fractional Negative Numbers: Appear when the greater number is subtracted from the smaller one: also appear as a result division of a negative integer by a natural number
• Positive Numbers: Numbers (integers and fractional ones), considered in arithmetic.
• Rational Numbers: Positive and negative numbers (integers and fractional ones) and zero.

Irrational numbers

• Irrational Numbers: NOT represented as a vulgar fraction; numbers are calculated with any accuracy, but can’t be changed by a rational number; appear as results of geometrical measurements, for example: a ratio of a square diagonal length to its side length is equal to √2, a ratio of a circumference length to its diameter length is an irrational number π;
• Real Numbers: Representable by an infinite decimal expansion, which may be repeating or nonrepeating; in a one-to-one correspondence with the points on a straight line; that have a nonrepeating decimal expansion are called irrational, i.e., they cannot be represented by any ratio of integers.

Imaginary and complex numbers

• Imaginary Numbers: Solution of x² = a at negative values of a, the new kind numbers; the second power is a negative number; an imaginary unit is defined as i = √-1;
• All the rest are real numbers and a sum of a real and an imaginary number result in a Complex Numbers, and is marked as: a+bi

Monomials and Polynomials

• Monomial: expression that is a product of numbers, variables, and/or powers of variables.
  • A single number or a single letter may be also considered as a monomial
  • Any factor of a monomial may be called a coefficient
  • Monomials are called similar or like ones, if they are identical or differed only by coefficients
  • Degree of monomial is a sum of exponents of the powers of all its letters
• Addition of monomials: If among a sum of monomials there are similar ones, he sum can be reduced to the more simple form in which exponents of the powers are added
• Multiplication of monomials: A product of some monomials can be simplified, only if it has powers of the same letters or numerical coefficients. In this case exponents of the powers are added and numerical coefficients are multiplied.
• Polynomial: algebraic sum of monomials -Degree of polynomial is the most of degrees of monomials, forming this polynomial
  • Multiplication of sums and polynomials: a product of the sum of two or some expressions by any expression is equal to the sum of the products of each of the addends by this expression
• Algebraic Fraction: expression of a shape A/B, where A and B can be a number, a monomial, a polynomial. As in arithmetic, A is called a numerator, B is a denominator and a cancellation can be made depending on the operations being done on the fraction