NES Elementary Education Subtest 2 complete solution updated 2024/2025 Natural Numbers - N = {1, 2, 3, 4, 5, 6, . . . } Whole natural numbers together with zero. - W = {0, 1, 2, 3, 4, 5, 6, . . . } Every whole number has a unique opposite or ne

NES Elementary Education Subtest 2 complete solution updated 2024/2025 Natural Numbers - N = {1, 2, 3, 4, 5, 6, . . . } Whole natural numbers together with zero. - W = {0, 1, 2, 3, 4, 5, 6, . . . } Every whole number has a unique opposite or negative whose sum with it is 0. For example, - 2 + (-2) = 0 The set of integers consists of the whole numbers and their opposites. - Z = {. . ., -3, -2, -1, 0, 1, 2, 3, . . . } Every nonzero integer has a unique reciprocal whose product with it is one. For example, - 2 × 1/2 = 1 The ratio or fraction of one integer to a nonzero integer is the product of the first integer with the reciprocal of the second. For example, the ratio of 2 to 3 is - 2/3 = 2 × 1/3 not every rational number is an integer. For example, 1/2 is a rational number that is not an integer. - 1/2 = 0.5 There are three basic properties of addition: - commutativity, associativity and identity. Commutative property. - When adding two numbers, the sum is the same regardless of the order in which the numbers are added. 2 + 3 = 3 + 2 Associative property. - When adding three or more numbers, the sum is the same regardless of the way in which the numbers are grouped. 2 + (3 + 5) = (2 + 3) + 5 Identity property. - Adding zero to a number does not change it. 2 + 0 = 2 There are three basic properties of multiplication: - commutativity, associativity and identity. Distributive property. - The product of a number with a sum equals the sum of the products of the number with each term of the sum. 2 × (3 + 5) = (2 × 3) + (2 × 5) Exponentiation - Exponentiation is repeated multiplication. An exponent is often called a power. For example, the third power of 2 is 2³ = 2 × 2 × 2 = 8 We define the zero power of any nonzero number to be 1. For example, - (-3)0 = 1 A negative exponent indicates a reciprocal. For example, - 2 (-3rd power) =1 / 2 (3rd power) = 1 / 8 The first power of any number is itself. For example, - 2 (to the 1st power) = 2 To multiply like bases with exponents, add the exponents. For example, - 2 (to the 3rd) x 2 (to the 5th) = 2 (to the eighth) To exponentiate a power, multiply the exponents. For example, - (2 to the 3rd) to the 5th = 2 to the 15th 10 to the 0 power - 1 10 to the 1 power - 10 10 to the -2 power - 1 / 10 to the 2 power or 1 / 100 10 to the 2 power x 10 to the 3 power - 10 to the 5 power Identifying Place Value in Numbers = (2 x 10 to the 3 power) + (0 x 10 to the 2 power) + (4 x 10 to the 1 power) + 5 x 10 to the 0 power) Digits to the right of a decimal point correspond to negative powers of ten. For example, 23.405 - 23.405 = (2 x 10 to the 1 power) + (3 x 10 to the 0 power) + (4 x 10 to the -1 power) + (0 x 10 to the -2 power) + (5 x 10 to the -3 power) Converting a fraction to a decimal. For example 3/8 - 3 divided by 8 = 0.375 convert 0.45 to a fraction. - 45/100 convert 3.208 to a mixed number. - 3 + 208/1000 Converting a fraction to a percentage. Convert the fraction to a decimal and then convert the decimal to a percentage. For example, 2/5 - 2/5 = .4 = 40% Converting a percentage to a fraction. Convert the percentage to fraction with a denominator of 100. For example, 65% - 65/100 A fundamental concept of mathematics is that the set of real numbers is in one-to-one correspondence with the set of points on a line. That is, each real number corresponds to exactly one point on a line, and each point on a line corresponds to exactly one real number, called the _____ of the point - coordinate It is worthwhile to memorize the first several prime numbers. - 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, ... A natural number is _____ if it is greater than 1 and not prime. - composite Divisibility Tests - To find the prime factorization of a number, it is helpful to know a few tests for divisibility. Divisibility by 2. - If the last digit is even, then the number is divisible by 2. For example, 158 is divisible by 2 since its last digit is 8. Divisibility by 3. - If the sum of the digits is divisible by 3, then the number is also. For example, 177 is divisible by 3 since the sum of its digits is 15 (1 + 7 + 7 = 15), and 15 is divisible by 3. Divisibility by 4. - If the last two digits form a number divisible by 4, then the number is divisible by 4. For example, 316 is divisible by 4 since 16 is divisible by 4. Divisibility by 5. - If the last digit is a 5 or a 0, then the number is divisible by 5. For example, 1995 is divisible by 5 since its last digit is 5. Divisibility by 6. - If the number is divisible by both 3 and 2, then it is also divisible by 6. For example, 168 is divisible by 6 since it is divisible by 2, and it is divisible by 3. Divisibility by 8. - If the last three digits form a number divisible by 8, then the number itself is also divisible by 8. For example, 1,120 is divisible by 8 since 120 is divisible by 8. Divisibility by 9. - If the sum of the digits is divisible by 9, then the number itself is also divisible by 9. For example, 369 is divisible by 9 since the sum of its digits is 18 (3 + 6 + 9 = 18), and 18 is divisible by 9. Divisibility by 10. - If the number ends in 0, then it is divisible by 10. For example, 670 is divisible by 10 since its last digit is 0. The Fibonacci Sequence. The Fibonacci sequence is the sequence - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...