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Introduction to Arithmetic
Hey there, future mathematicians! Arithmetic is a super important part of math. It's all about numbers and how we play with them. Think of it as the foundation for all the cool math stuff you'll learn later, like algebra, geometry, and calculus. So, let's make sure we get it right!
The Basic Operations
1. Addition (+)
Addition is putting things together. We use the symbol "+" to show addition. โ Example 1: You have 3 apples, and your friend gives you 2 more. How many apples do you have? โ Answer: 3 + 2 = 5 apples โ Example 2: What is the sum of 124 and 25? โ Answer: 124 + 25 = 149
Properties of Addition
Understanding these properties can make addition easier and faster! โ Commutative Property: The order in which you add numbers doesn't change the result. โ a + b = b + a โ Example: 2 + 3 = 3 + 2 = 5 โ It's like saying whether you put the apples in the basket first or the oranges, you'll still have the same total amount of fruit. โ Associative Property: When you add three or more numbers, the way you group them doesn't change the result. โ (a + b) + c = a + (b + c) โ Example: (1 + 2) + 3 = 1 + (2 + 3) = 6 โ This means if you're adding a few numbers, you can add any two of them together first. It won't change your final answer. โ Identity Property: Adding zero to any number doesn't change the number. Zero is Zero is the "additive identity." โ a + 0 = a โ Example: 7 + 0 = 7 โ Zero is like the special number that doesn't change anything when you add it to another number.
2. Subtraction (-)
Subtraction is taking things away. We use the symbol "-" to show subtraction. โ Example 1: You have 7 cookies, and you eat 2. How many cookies are left? โ Answer: 7 - 2 = 5 cookies โ Example 2: What is 235 minus 112? โ Answer: 235 - 112 = 123
3. Multiplication (x or *)
Multiplication is like repeated addition. We use the symbols "x" or "*" to show multiplication. โ Example 1: You have 3 bags of candy, and each bag has 4 candies. How many candies do you have? โ Answer: 3 x 4 = 12 candies โ Example 2: What is 15 multiplied by 6? โ Answer: 15 x 6 = 90
Properties of Multiplication
These properties make multiplication easier to work with. โ Commutative Property: The order in which you multiply numbers doesn't change the result. โ a x b = b x a โ Example: 2 x 3 = 3 x 2 = 6 โ Just like addition, you can multiply numbers in any order. โ Associative Property: When you multiply three or more numbers, the way you group them doesn't change the result. โ (a x b) x c = a x (b x c) โ Example: (2 x 3) x 4 = 2 x (3 x 4) = 24 โ When you have a series of multiplications, you can group any pair of numbers together to multiply first. โ Identity Property: Multiplying any number by one doesn't change the number. One is the "multiplicative identity." โ a x 1 = a โ Example: 9 x 1 = 9 โ One is the special number that leaves every other number unchanged when you multiply by it. โ Zero Property: Multiplying any number by zero always gives zero. โ a x 0 = 0 โ Example: 5 x 0 = 0 โ Zero has a powerful effect in multiplication: it always results in zero. โ Distributive Property: This property combines multiplication and addition (or subtraction). It states that multiplying a sum (or difference) by a number is the same as multiplying each term of the sum (or difference) by that number and then adding (or subtracting) the products.
denominator (the bottom number). โ Equivalent Fractions: Fractions that represent the same value, but have different numerators and denominators. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. โ Example: 1/2 = 2/4 = 3/6 = 4/ โ Simplifying Fractions (Reducing to Lowest Terms): Dividing the numerator and denominator by their greatest common factor (GCF) to get the simplest form of the fraction. โ Example: 6/8. The GCF of 6 and 8 is 2. So, 6/8 simplified is (6รท2)/(8รท2) = 3/ โ Adding and Subtracting Fractions: โ If the denominators are the same, add or subtract the numerators and keep the denominator. โ Example: 1/5 + 2/5 = (1+2)/5 = 3/ โ If the denominators are different, find a common denominator (usually the least common multiple - LCM), create equivalent fractions, and then add or subtract. โ Example: 1/2 + 1/3. The LCM of 2 and 3 is 6. So, 1/2 = 3/6 and 1/3 = 2/6. Therefore, 1/2 + 1/3 = 3/6 + 2/6 = 5/ โ Multiplying Fractions: Multiply the numerators and multiply the denominators. โ Example: (2/3) x (4/5) = (2x4)/(3x5) = 8/ โ Dividing Fractions: Multiply the first fraction by the reciprocal of the second fraction (flip the second fraction). โ Example: (1/2) รท (3/4) = (1/2) x (4/3) = (1x4)/(2x3) = 4/6 = 2/ โ Example 1: You eat 1/2 of a pizza. How much of the pizza is left? โ Answer: 1 - 1/2 = 2/2 - 1/2 = 1/ โ Example 2: What is 1/4 of 12? โ Answer: (1/4) x 12 = (1/4) x (12/1) = 12/4 = 3
Decimals
Decimals are another way to represent parts of a whole. They are based on powers of 10. Each digit in a decimal has a place value (tenths, hundredths, thousandths, etc.). โ Adding and Subtracting Decimals: Line up the decimal points, add or subtract as you would with whole numbers, and place the decimal point in the answer directly below the decimal points in the problem. โ Example: 12.34 + 5.67 = 18. โ Example: 9.87 - 4.56 = 5. โ Multiplying Decimals: Multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the factors and place the decimal point in the product so that it has that many decimal places. โ Example: 2.5 x 3.2 = 8.00 (2 decimal places in the answer) โ Dividing Decimals: โ If the divisor is a whole number, divide as you would with whole numbers. Place
the decimal point in the quotient directly above the decimal point in the dividend. โ Example: 7.2 รท 3 = 2. โ If the divisor is a decimal, move the decimal point in the divisor to the right until it becomes a whole number. Move the decimal point in the dividend the same number of places to the right. Then divide. โ Example: 12.4 รท 0.2 = 124 รท 2 = 62 โ Converting between Decimals and Fractions: โ Decimal to Fraction: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., depending on the number of decimal places. Then simplify the fraction. โ Example: 0.75 = 75/100 = 3/ โ Fraction to Decimal: Divide the numerator by the denominator. โ Example: 1/4 = 1 รท 4 = 0. โ Example 1: You have $5.75, and you spend $2.50. How much money do you have left? โ Answer: $5.75 - $2.50 = $3. โ Example 2: What is 0.5 multiplied by 10? โ Answer: 0.5 x 10 = 5
Percentages
Percentages are a way to express a number as a fraction of 100. The word "percent" means "per hundred." The symbol for percent is %. โ Converting between Percentages, Fractions, and Decimals: โ Percentage to Decimal: Divide the percentage by 100 and remove the % sign. โ Example: 50% = 50/100 = 0.50 = 0. โ Decimal to Percentage: Multiply the decimal by 100 and add the % sign. โ Example: 0.25 = 0.25 x 100% = 25% โ Percentage to Fraction: Write the percentage as a fraction with a denominator of 100 and simplify. โ Example: 75% = 75/100 = 3/ โ Fraction to Percentage: Convert the fraction to a decimal by dividing the numerator by the denominator, then multiply by 100% โ Example: 1/5 = 0.2 = 0.2 x 100% = 20% โ Finding the Percentage of a Number: Multiply the number by the percentage (expressed as a decimal or fraction). โ Example: What is 20% of 60? 20% of 60 = 0.20 x 60 = 12 or 20% of 60 = (20/100) x 60 = 12 โ Finding What Percentage One Number is of Another: Divide the first number by the second number and multiply by 100%. โ Example: What percentage is 15 of 75? (15/75) x 100% = 0.20 x 100% = 20% โ Finding the Whole When a Percentage is Known: Divide the known part by the percentage (expressed as a decimal or fraction).
โ Example: (-3) + (-2) = - โ Example: 4 + 5 = 9 โ If the signs are different, subtract the smaller number from the larger number and take the sign of the larger number. โ Example: (-5) + 2 = - โ Example: 7 + (-3) = 4 โ Subtracting Integers: โ To subtract an integer, add its opposite. โ Example: 4 - (-3) = 4 + 3 = 7 โ Example: -2 - 5 = -2 + (-5) = - โ Example: -6 - (-1) = -6 + 1 = - โ Multiplying and Dividing Integers: โ If the signs are the same, the result is positive. โ Example: (-2) x (-4) = 8 โ Example: 3 x 5 = 15 โ Example: (-6) รท (-2) = 3 โ Example: 8 รท 4 = 2 โ If the signs are different, the result is negative. โ Example: (-6) x 3 = - โ Example: 4 x (-2) = - โ Example: 10 รท (-2) = - โ Example: (-12) รท 4 = -
Exponents
Exponents indicate how many times a base number is multiplied by itself. For example, in 2^3, 2 is the base and 3 is the exponent, and it means 2 x 2 x 2 = 8. โ Terminology: โ Base: The number being multiplied. โ Exponent (or Power): The number that indicates how many times the base is multiplied by itself. โ Examples: โ 5^2 (5 to the power of 2, or 5 squared) = 5 x 5 = 25 โ 2^4 (2 to the power of 4) = 2 x 2 x 2 x 2 = 16 โ 10^3 (10 to the power of 3, or 10 cubed) = 10 x 10 x 10 = 1000 โ Rules of Exponents: โ Product Rule: a^m x a^n = a^(m+n) (When multiplying powers with the same base, add the exponents) โ Example: 2^2 x 2^3 = 2^(2+3) = 2^5 = 32 โ Quotient Rule: a^m รท a^n = a^(m-n) (When dividing powers with the same base, subtract the exponents) โ Example: 5^4 รท 5^2 = 5^(4-2) = 5^2 = 25
โ Power of a Power Rule: (a^m)^n = a^(m x n) (When raising a power to a power, multiply the exponents) โ Example: (3^2)^3 = 3^(2x3) = 3^6 = 729 โ Power of a Product Rule: (a x b)^m = a^m x b^m (When raising a product to a power, distribute the power to each factor) โ Example: (2 x 3)^2 = 2^2 x 3^2 = 4 x 9 = 36 โ Power of a Quotient Rule: (a/b)^m = a^m / b^m (When raising a quotient to a power, distribute the power to both the numerator and denominator) โ Example: (4/2)^3 = 4^3 / 2^3 = 64 / 8 = 8 โ Zero Exponent Rule: a^0 = 1 (Any non-zero number raised to the power of 0 is 1) โ Example: 7^0 = 1 โ Negative Exponent Rule: a^(-n) = 1/a^n (A negative exponent means taking the reciprocal of the base raised to the positive exponent) โ Example: 2^(-3) = 1/2^3 = 1/
Roots
A root is the opposite of an exponent. The most common root is the square root, which finds a number that, when multiplied by itself, equals a given number. For example, the square root of 9 is 3 because 3 x 3 = 9. โ Terminology: โ Radical Symbol: โ (This symbol is used to indicate a root) โ Index: The small number written above and to the left of the radical symbol, indicating the type of root (e.g., 2 for square root, 3 for cube root). If no index is written, it is assumed to be 2 (square root). โ Radicand: The number under the radical symbol. โ Examples: โ โ9 (square root of 9) = 3, because 3 x 3 = 9 โ โ8 (cube root of 8) = 2, because 2 x 2 x 2 = 8 โ โ25 (square root of 25) = 5, because 5 x 5 = 25 โ โ16 (fourth root of 16) = 2, because 2 x 2 x 2 x 2 = 16 โ Simplifying Roots: โ Find the largest perfect square (for square roots), perfect cube (for cube roots), etc., that is a factor of the radicand. โ Use the property โ(a x b) = โa x โb (and the equivalent for cube roots, etc.) to separate the root into the product of the root of the perfect power and the root of the remaining factor. โ Simplify the root of the perfect power. โ Example: โ32 = โ(16 x 2) = โ16 x โ2 = 4โ โ Example: โ24 = โ(8 x 3) = โ8 x โ3 = 2 โ 3
Ratios and Proportions
