AMT111 Chapter 1 - Arithmetic and Number Systems PDF

# Section A: Arithmetic ## Introduction * Arithmetic is the most basic branch of mathematics. * It uses real, non-negative numbers (counting numbers). * Arithmetic consists of four operations: addition, subtraction, multiplication, and division. ## Number Systems * **Decimal System (Base Ten):**...

# Section A: Arithmetic ## Introduction * Arithmetic is the most basic branch of mathematics. * It uses real, non-negative numbers (counting numbers). * Arithmetic consists of four operations: addition, subtraction, multiplication, and division. ## Number Systems * **Decimal System (Base Ten):** * Uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. * Digits are reused in various combinations to represent larger numbers. * Numbers are arranged in columns based on multiples of ten. * Uses fractions and decimals to represent quantities between whole numbers. * **Binary System (Base Two):** * Uses only two digits: 0 and 1. * Represents numbers using combinations of ON and OFF electrical signals. * Used in computer calculations. ## Whole Numbers * **Operations:** addition, subtraction, multiplication, and division. * **Addition:** * Process of finding the total of two or more numbers. * Indicated by the plus (+) sign. * The result is called the sum. * **Subtraction:** * Process of finding the difference between two numbers. * Indicated by the minus (-) sign. * The number being subtracted is the subtrahend. * The number being subtracted from is the minuend. * **Multiplication:** * Special form of repetitive addition. * Indicated by (×), (•), or lack of other operation signs. * The number being multiplied is the multiplicand. * The number indicating how many times to multiply is the multiplier. * **Division:** * The reverse of multiplication. * Indicated by the division sign (+), or a with the dividend inside and the divisor to the left. * The number being divided is the dividend. * The number dividing is the divisor. * The result is the quotient. * May include a remainder. ## Signed Numbers * Numbers larger than zero have a positive value. * Numbers smaller than zero have a negative value. ## Adding Signed Numbers * **Same Signs:** Ignore the sign, add the values, and place the common sign in front of the answer. * **Different Signs:** Find the difference between the values, and apply the sign of the larger number. ## Subtracting Signed Numbers * Change the operation sign to plus and change the sign of the subtrahend. * Proceed as you would in addition. ## Multiplying Signed Numbers * **Positive and Positive:** The product is always positive. * **Negative and Negative:** The product is always positive. * **Positive and Negative:** The product is always negative. ## Dividing Signed Numbers * **Positive and Positive:** The quotient is always positive. * **Negative and Negative:** The quotient is always positive. * **Positive and Negative:** The quotient is always negative. ## Common Fractions * Represents a portion or part of a quantity. * Consists of a numerator and a denominator. * **Proper Fraction:** Numerator is smaller than the denominator. The fraction is less than 1. * **Improper Fraction:** Numerator is larger than the denominator. The fraction is greater than 1. * **Mixed Number:** Combination of a whole number and a proper fraction. ## Reducing Fractions * Simplifying a fraction to its lowest terms. * Divide both the numerator and denominator by their greatest common factor. ## Adding Common Fractions * Find the least common denominator (LCD). * Add the numerators. * Express the answer over the LCD and reduce. ## Subtracting Common Fractions * Find the LCD. * Subtract the numerators. * Express the answer over the LCD and reduce. ## Mixed Numbers * Combination of a whole number and a proper fraction. * **Converting to an Improper Fraction:** Multiply the whole number by the denominator, add the numerator, and place the sum over the original denominator. * **Adding:** * Convert mixed numbers to improper fractions. * Find the LCD. * Add the numerators. * Reduce the answer and convert to a mixed number if necessary. * **Subtracting:** * Convert mixed numbers to improper fractions. * Find the LCD. * Subtract the numerators. * Reduce the answer and convert to a mixed number if necessary. ## Multiplying Fractions * **Multiply the Numerators:** Multiply the numerators of the fractions to get the product numerator. * **Multiply the Denominators:** Multiply the denominators of the fractions to get the product denominator. * **Reduce the Fraction:** Simplify the resulting fraction to its lowest terms. ## Dividing Fractions * **Invert the Divisor:** Flip the divisor (the fraction after the division sign) so that the numerator becomes the denominator and vice versa. * **Multiply:** Multiply the dividend by the inverted divisor. ## Decimals * Fractions where the denominator is a power of ten (10, 100, 1000, etc.). * The position of each digit to the right of the decimal point indicates a fraction. ## Adding Decimals * Align the decimal points vertically. * Add as you would with whole numbers. * Place the decimal point in the answer directly below the decimal points in the numbers you added. ## Subtracting Decimals * Align the decimal points vertically. * Subtract as you would with whole numbers. * Place the decimal point in the answer directly below the decimal points in the numbers you subtracted. ## Multiplying Decimals * Ignore the decimal points and multiply the whole numbers. * Count the total number of digits to the right of the decimal point in both the multiplier and multiplicand. * Place the decimal point in the product that many places from the right. ## Dividing Decimals * **Whole Number Divisor:** Align the decimal point in the quotient directly below the decimal point in the dividend. * **Decimal Divisor:** * Convert the divisor to a whole number by moving the decimal point to the right. * Move the decimal point in the dividend the same number of places to the right. * Divide as usual. ## Converting Decimals to Fractions 1. Write the decimal as a fraction with the decimal number as the numerator and 1 followed by the same number of zeros as decimal places as the denominator. 2. Simplify the fraction to its lowest terms. ## Rounding Decimals * **Identify the last retained digit:** The digit you want to keep. * **Look at the next digit:** The digit immediately to the right of the last retained digit. * **Round up:** If the next digit is 5 or greater, increase the last retained digit by 1. * **Leave unchanged:** If the next digit is less than 5, leave the last retained digit unchanged. ## Percentage * A fraction whose denominator is 100. * Represented by the symbol %. ## Ratio and Proportion * **Ratio:** A way to compare two quantities. * Written as a:b, a/b, or a to b. * **Proportion:** A statement of equality between two ratios. * Written as a:b = c:d or a/b = c/d. ## Powers and Roots * **Power:** A number multiplied by itself a certain number of times. * Written as a^n, where a is the base number and n is the exponent. * The exponent indicates how many times the base is multiplied by itself * **Root:** A number that, when multiplied by itself a certain number of times, equals a given number. * The square root of a number is the root when multiplied by itself once. * The cube root of a number is the root when multiplied by itself twice. ## Scientific Notation * A way to write very large or very small numbers concisely. * Written as a x 10^n, where: * a is a number between 1 and 10. * n is an integer, representing the power of ten. * **Multiplication / Division:** * Multiply the numbers as usual and add/subtract the exponents. ## Division Using Scientific Notation * Convert the numbers to scientific notation. * Divide the numbers as usual and subtract the exponents.