Tiêu đề Math Cheat Sheet.pdf ✅

A) Elementary Math: Arithmetic and pre-algebra If you’re like me, you probably haven’t paid much serious attention to basic topics since junior high school. You’ll need to learn about them again if you want to do well on the test. By the time you take the exam, using them should be automatic. All the arithmetic concepts are fairly basic, but you’ll have to know them well. You should know the following arithmetic terms: a) The result of addition or adding expressions is called a sum or total. b) The result of subtraction is called a difference. c) The result of multiplication is called a product. d) The result of division is called a quotient. e) In the expression 52, the 2 is called an exponent. ORDER OF OPERATIONS Oftentimes, solving an equation on the test will require you to perform several different operations, one after the other. These operations must be performed in the proper order. In general, the problems are written in such a way that you won’t have trouble deciding what comes first. In cases in which you are uncertain, you need to remember only the following sentence: Please Excuse My Dear Aunt Sally; she limps from left to right. That’s PEMDAS, for short and It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. First, do any calculations inside the parentheses; then take care of the exponents; then perform all multiplication and division, from left to right, followed by addition and subtraction, from left to right. Majority of the time, you may not have to deal with PEMDAS. Just follow the symbols of grouping below, in order of priorities. a) Solve the operations inside the parentheses ( ) first, then b) solve for operations inside the brackets [ ] then finally, c) solve for operations inside the braces { }. Also, take note of the following Properties for addition/subtraction: a) Commutative Property: for any number and , examples: is the same as 2 + and the same as -4 + 7 is the same as 7 + or b) Associative Property: for any number and , examples: 5 + (3 - 4) = 5 + (3 + (-4)) = (5+3) + (-4) = c) Additive inverse: for any number a, the additive inverse is the negative of . a b a + b = b + a 2 − 3 (−3) −3 + 2 (−3) 7 − 3 a, b c a + (b + c) = (a + b) + c (5 + 3) − 4 (−5) + (3 + 2) = ((−5) + 3) + 2 a Page 1 of 52
example: the additive inverse of 5 is . For multiplication /division a) Commutative Property: for any number and , examples: b) Associative Property: for any number and , example: c) Multiplicative inverse: for any number , the multiplicative inverse is example: the multiplicative inverse of 7 is the multiplicative inverse of is d) Distributive Property 1) Left hand distributive: examples: 2) right hand distributive: examples: Some helpful theorems: (we often neglect) 1) If , then Interpretation: when you move a number to the opposite side of the equation, you change the sign of that number. example: If , then 2) For any real number , Interpretation: the negative of a negative number is positive example: 3) For any real number , Interpretation: any number multiplied by zero is zero examples: , or 4) if and only if or −5 a b a ⋅ b = b ⋅ a 3 ⋅ 4 = 4 ⋅ 3 2 ⋅ (−3) = (−3) ⋅ 2 a, b c a(bc) = (ab)c 2(3 ⋅ (−4)) = (2 ⋅ 3)(−4) a 1 a 1 7 −4 −1 4 a(b + c) = ab + ac 4(3 + 8) = 4 ⋅ 3 + 4 ⋅ 8 3(4 − 9) = 3 ⋅ 4 − 3 ⋅ 9 −5(−3 + 4) = (−5)(−3) + (−5) ⋅ 4 (a + b)c = ac + bc (7 + 6)8 = 7 ⋅ 8 + 6 ⋅ 8 (4 − 7)(−5) = 4(−5) − 7(−5) (−5 + 6)9 = (−5)9 + 6 ⋅ 9 a + b = 0 a = − b x + 3 = 0 x = − 3 a −(−a) = a −(−24) = 24 a a ⋅ 0 = 0 5 ⋅ 0 = 0 0 ⋅ (−100) = 0 ab = 0 a = 0 b = 0 Page 2 of 52
Interpretation: the product of two numbers is zero when one of the numbers is zero. (or both numbers is zero) 5) Interpretation: the negative sign is not distributive over multiplication. Its like multiplying numbers by - 1 Example: 6) Interpretation: division is the same as multiplication by the reciprocal Example: (note: is the reciprocal of 5) 7) Interpretation: negative sign is distributive over addition Example: 8) If , then Interpretation: If the bottom of two fractions is the same, you can simply add the top and copy the bottom Example: 9) Interpretation: In a fraction, the negative sign can be placed in the middle, top to bottom. The value of the fraction is still negative. 10) Interpretation: Dividing by fractions is the same as multiplying by the reciprocal of the bottom. 11) If , then Interpretation: We call this cross multiplication in Math class DIVISIBILITY RULES 1 1) A number is divisible by 2 if the last digit is even. (−a)b = − (ab) (−50)(23) = − (50 ⋅ 23) a ÷ b = a ⋅ 1 b 3 ÷ 5 = 3 ⋅ 1 5 1 5 −(a + b) = − a − b −(3 + 4) = − 3 − 4 c ≠ 0 a c + b c = a + b c 2 7 + 4 7 = 6 7 −a b = − a b = a −b a b c d = a b ⋅ d c a b = c d ad = bc Source: https://www.mathsisfun.com/divisibility-rules.html 1 Page 3 of 52
example: 156 is is divisible by 2 and 135 is not. 2) A number is divisible by 3 if the sum of the digits is divisible by 3 example: 381 (3+8+1 = 12 3 = 4) Yes! 217 (2+1+7 = 10 3 = 3.333) No! 3) A number is divisible by 4 if the last 2 digits are divisible by 4 example: 1312 is (12÷4=3) Yes 7019 is not (19÷4=4 3/4) No 4) A number is divisible by 5 if the last digit is 0 or 5 example: 185 is divisible by 5 and 213 is not A quick check (useful for small numbers) is to halve the number twice and the result is still a whole number. example: 12/2 = 6, 6/2 = 3, 3 is a whole number. Yes 30/2 = 15, 15/2 = 7.5 which is not a whole number. No 5) A number is divisible by 6 if the number is even and is divisible by 3 example: 114 (it is even, and 1+1+4=6 and 6÷3 = 2) Yes 308 (it is even, but 3+0+8=11 and 11÷3 = 3 2/3) No 6) A number is divisible by 8 if the last three digits are divisible by 8 example: 109816 (816÷8=102) Yes 216302 (302÷8=37 3/4) No A quick check is to halve three times and the result is still a whole number: 816/2 = 408, 408/2 = 204, 204/2 = 102 Yes 302/2 = 151, 151/2 = 75.5 No 7) A number is divisible by 9 if the sum of the digits is divisible by 9 example: 1629 (1+6+2+9=18, and again, 1+8=9) Yes 2013 (2+0+1+3=6) No 8) A number is divisible by 10 if the number ends in 0 Example: 220 is divisible by 10 while 221 was not Some terminologies on the set of real numbers 1) Counting numbers - or Natural numbers, are numbers 1,2,3 ... and so on. These numbers are called counting numbers because we learn this as we start to "count" ÷ ÷ Page 4 of 52