Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
The Basic Operations and Special Properties http://www.youtube.com/watch?v=MOYNs-FXKfY In basic mathematics there are many ways of saying the same thing: Symbol + Words Used Addition, Add, Sum, Plus, Increase, Total - Subtraction, Subtract, Minus, Less, Difference, Decrease, Take Away, Deduct × Multiplication, Multiply, Product, By, Times, Lots Of ÷ Division, Divide, Quotient, Goes Into, How Many Times Addition is ... ... bringing two or more numbers (or things) together to make a new total. The numbers to be added together are called the "Addends": Subtraction is ...taking one number away from another. Minuend - Subtrahend = Difference Minuend: The number that is to be subtracted from. Subtrahend: The number that is to be subtracted. Difference: The result of subtracting one number from another. Multiplication is ... .. (in its simplest form) repeated addition. Here we see that 6+6+6 (three 6s) make 18 It could also be said that 3+3+3+3+3+3 (six 3s) make 18 Division is ... ... splitting into equal parts or groups. It is the result of "fair sharing". Division has its own special words to remember. Let's take the simple problem of dividing 22 by 5. The answer is 4, with 2 left over. Here we illustrate the important words: Which is the same as: Average You calculate the average by adding up all the values, then divide by how many values. Example: What is the average of 9, 2, 12 and 5? Add up all the values: 9 + 2 + 12 + 5 = 28 Divide by how many values (there are four of them): 28 ÷ 4 = 7 So the average is 7 This makes it the Additive Identity, which is just a special way of saying "add 0 and you get the identical number you started with". Special Properties http://www.youtube.com/watch?v=c3Z59HjNxEQ Additive Identity Adding zero to a number leaves it unchanged: a+0=0+a=a Additive Inverse What you add to a number to get zero. The negative of a number. Example: The additive inverse of -5 is 5, because -5 + 5 = 0. The additive inverse of +5 is -5 as well. Multiplicative Inverse Another name for Reciprocal. When you multiply a number by its "Multiplicative Inverse" you get 1. Example: 8 × (1/8) = 1 Associative Law The "Associative Laws" say: * It doesn't matter how you group the numbers when you add. * It doesn't matter how you group the numbers when you multiply. (In other words it doesn't matter which you calculate first.) Example addition: (6 + 3) + 4 = 6 + (3 + 4) Because 9 + 4 = 6 + 7 = 13 Example multiplication: (2 × 4) × 3 = 2 × (4 × 3) ' cause 8 × 3 = 2 × 12 = 24 Commutative, Associative and Distributive Laws Wow! What a mouthful of words! But the ideas are simple. The "Commutative Laws" say you can swap numbers over and still get the same answer .... when you add: a+b = b+a Example: ... or when you multiply: a×b = b×a Example: Associative Laws The "Associative Laws" say that it doesn't matter how you group the numbers (i.e. which you calculate first) ... ... when you add: (a + b) + c = a + (b + c) ... or when you multiply: (a × b) × c = a × (b × c) Examples: This: (2 + 4) + 5 = 6 + 5 = 11 Has the same answer as this: 2 + (4 + 5) = 2 + 9 = 11 This: (3 × 4) × 5 = 12 × 5 = 60 Has the same answer as this: 3 × (4 × 5) = 3 × 20 = 60 Uses: Sometimes it is easier to add or multiply in a different order: What is 19 + 36 + 4? 19 + 36 + 4 = 19 + (36 + 4) = 19 + 40 = 59 Or to rearrange a little: What is 2 × 16 × 5? 2 × 16 × 5 = (2 × 5) × 16 = 10 × 16 = 160 Distributive Law The "Distributive Law" is the BEST one of all, but needs careful attention. This is what it lets you do: 3 lots of (2+4) is the same as 3 lots of 2 plus 3 lots of 4 So, the 3× can be "distributed" across the 2+4, into 3×2 and 3×4 And we write it like this: a × (b + c) = a × b + a × c Try the calculations yourself: 3 × (2 + 4) = 3 × 6 = 18 3×2 + 3×4 = 6 + 12 = 18 Either way gets the same answer. In English we can say: You get the same answer when you: multiply a number by a group of numbers added together, or do each multiply separately then add them Uses: Sometimes it is easier to break up a difficult multiplication: Example: What is 6 × 204 ? 6 × 204 = 6×200 + 6×4 = 1,200 + 24 = 1,224 Or to combine: Example: What is 16 × 6 + 16 × 4? 16 × 6 + 16 × 4 = 16 × (6+4) = 16 × 10 = 160 You can use it in subtraction too: Example: 26×3 - 24×3 26×3 - 24×3 = (26 - 24) × 3 = 2 × 3 = 6 You could use it for a long list of additions, too: Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7 6×7 + 2×7 + 3×7 + 5×7 + 4×7 = (6+2+3+5+4) × 7 = 20 × 7 = 140 And those are the Laws! But Not ... But don't go too far! The Commutative Law does not work for division: Example: 12 / 3 = 4, but 3 / 12 = ¼ The Associative Law does not work for subtraction: Example: (9 – 4) – 3 = 5 – 3 = 2, but 9 – (4 – 3) = 9 – 1 = 8 The Distributive Law does not work for division: Example: 24 / (4 + 8) = 24 / 12 = 2, but 24 / 4 + 24 / 8 = 6 + 3 = 9 Summary Commutative Laws: a+b = b+a a×b = b×a Associative Laws: (a + b) + c = a + (b + c) (a × b) × c = a × (b × c) Distributive Law: a × (b + c) = a × b + a × c Expanded Notation Writing a number to show the value of each digit. It is shown as a sum of each digit multiplied by its matching place value (units, tens, hundreds, etc.) For example: 4,265 = 4 x 1,000 + 2 x 100 + 6 x 10 + 5 x 1 Here are some of zero's properties: Property Example a+0=a 4+0=4 a−0=a 4−0=4 a×0=0 6×0=0 0/a=0 0/3 = 0 a / 0 = undefined (dividing by zero is undefined) 7/0 = undefined 0a = 0 (a is positive) 04 = 0 00 = indeterminate 00 = indeterminate 0a = undefined (a is negative) 0-2 = undefined 0! = 1 ("!" is the factorial function) 0! = 1