1.6 multiplication i w

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Published on February 6, 2014

Author: Tzenma

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Multiplication I http://www.lahc.edu/math/frankma.htm

Multiplication I We simplify the notation for adding the same quantity repeatedly.

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2+2+2=6 as 3 x 2 or 3*2 or 3(2) = 6.

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”. Note that 3 copies so that 3 x 2 = 2 x 3. = 2 copies

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”. Note that 3 copies = 2 copies so that 3 x 2 = 2 x 3. In general, just as addition, multiplication is commutative, i.e. A x B = B x A.

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”. Note that 3 copies = 2 copies so that 3 x 2 = 2 x 3. In general, just as addition, multiplication is commutative, i.e. A x B = B x A. In the expression: 3 x 2 = 2 x 3 = 6

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”. Note that 3 copies = 2 copies so that 3 x 2 = 2 x 3. In general, just as addition, multiplication is commutative, i.e. A x B = B x A. In the expression: 3 x 2 = 2 x 3 = 6 the multiplicands 2 and 3 are called factors (of 6).

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”. Note that 3 copies = 2 copies so that 3 x 2 = 2 x 3. In general, just as addition, multiplication is commutative, i.e. A x B = B x A. In the expression: 3 x 2 = 2 x 3 = 6 the result 6 is called the product the multiplicands 2 and 3 (of 2 and 3). are called factors (of 6).

Multiplication I We simplify the notation for adding the same quantity repeatedly. For example, we shall write 3 copies 2 + 2 + 2 = 6 as 3 x 2 or 3*2 or 3(2) = 6. We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”. Note that 3 copies = 2 copies so that 3 x 2 = 2 x 3. In general, just as addition, multiplication is commutative, i.e. A x B = B x A. In the expression: 3 x 2 = 2 x 3 = 6 the result 6 is called the product the multiplicands 2 and 3 (of 2 and 3). are called factors (of 6). (Note: 1 and 6 are also factors of 6 because 1 x 6 = 6 x 1 = 6.)

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help.

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help. * (0 * x = 0 * x = 0) The product of zero with any number is 0.

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help. * (0 * x = 0 * x = 0) The product of zero with any number is 0. * (1 * x = x * 1 = x) The product of 1 with any number x is x.

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help. * (0 * x = 0 * x = 0) The product of zero with any number is 0. * (1 * x = x * 1 = x) The product of 1 with any number x is x. * For the products with 9 as a factor, the sum of their digits is 9.

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help. * (0 * x = 0 * x = 0) The product of zero with any number is 0. * (1 * x = x * 1 = x) The product of 1 with any number x is x. * For the products with 9 as a factor, the sum of their digits is 9. For example, 7 x 9 = 63 9 x 9 = 81 6 x 9 = 54 8 x 9 = 72 all have digit sum equal to 9,

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help. * (0 * x = 0 * x = 0) The product of zero with any number is 0. * (1 * x = x * 1 = x) The product of 1 with any number x is x. * For the products with 9 as a factor, the sum of their digits is 9. For example, 7 x 9 = 63 9 x 9 = 81 6 x 9 = 54 8 x 9 = 72 all have digit sum equal to 9, i.e. 5 + 4 = 9,

Multiplication I The multiplication table shown here is to be memorized and below are some features and tricks that might help. * (0 * x = 0 * x = 0) The product of zero with any number is 0. * (1 * x = x * 1 = x) The product of 1 with any number x is x. * For the products with 9 as a factor, the sum of their digits is 9. For example, 7 x 9 = 63 9 x 9 = 81 6 x 9 = 54 8 x 9 = 72 all have digit sum equal to 9, 7 + 2 = 9, 8+1=9 i.e. 5 + 4 = 9, 6+3=9

Multiplication I * The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table:

Multiplication I * The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table: 6636 6742 7749 6848 7856 8864 6954 7963 8972 9981

Multiplication I * The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table: 6636 6742 7749 6848 7856 8864 For example, 6 x 7 = 42 (= 7 x 6) 7 x 8 = 56 (= 8 x 7). 6954 7963 8972 9981

Multiplication I * The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table: 6636 6742 7749 6848 7856 8864 For example, 6954 7963 8972 9981 6 x 7 = 42 (= 7 x 6) 7 x 8 = 56 (= 8 x 7). The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etc are called even numbers.

Multiplication I * The following 4-digit numbers represent the products of the higher digits 6 thru 9, the more difficult part of the table: 6636 6742 7749 6848 7856 8864 For example, 6954 7963 8972 9981 6 x 7 = 42 (= 7 x 6) 7 x 8 = 56 (= 8 x 7). The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etc are called even numbers. The numbers 0(= 0*0), 1(= 1*1), 4(= 2*2), 9(= 3*3), 16(= 4*4),.., of the form x*x, down the diagonal, are called square numbers.

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done.

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number.

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, 7 x 4 7

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, record the unit-digit of the product, and carry the 10’s digit of the product. 7 x 4 7

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, record the unit-digit of the product, i. 4x7=28 and carry the 10’s digit of the product. 7 x 4 7

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, carry the 2 record the unit-digit of the product, i. 4x7=28 and carry the 10’s digit of the product. 7 x 4 7 8 record the 8,

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, carry the 2 record the unit-digit of the product, i. 4x7=28 and carry the 10’s digit of the product. ii. Multiply the next digit of the double 7 4 digit number to the single digit, x 7 8 record the 8,

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, carry the 2 record the unit-digit of the product, ii. 7x7=49, i. 4x7=28 and carry the 10’s digit of the product. ii. Multiply the next digit of the double 7 4 digit number to the single digit, x 7 8 record the 8,

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, carry the 2 record the unit-digit of the product, ii. 7x7=49, i. 4x7=28 and carry the 10’s digit of the product. 49+2=51 ii. Multiply the next digit of the double 7 4 digit number to the single digit, add the previous carry to the product, x 7 8 record the 8,

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, carry the 2 record the unit-digit of the product, ii. 7x7=49, i. 4x7=28 and carry the 10’s digit of the product. 49+2=51 ii. Multiply the next digit of the double 7 4 digit number to the single digit, add the previous carry to the product, x 7 record the unit-digit of this sum and 8 1 5 carry the 10’s digit of this sum. carry the 5 record the 1, record the 8,

Multiplication I The Vertical Format We use a vertical format to multiply larger numbers. The following demonstrates how this is done. We start with a two-digit number times a single digit number. For example, i. Starting from the right, multiply the two unit-digits, carry the 2 record the unit-digit of the product, ii. 7x7=49, i. 4x7=28 and carry the 10’s digit of the product. 49+2=51 ii. Multiply the next digit of the double 7 4 digit number to the single digit, add the previous carry to the product, x 7 record the unit-digit of this sum and 8 1 5 carry the 10’s digit of this sum. To multiply a longer number against a record carry record single digit number, repeat step ii until the 8, the 5 the 1, all the digits are multiplied.

Multiplication I Let’s add another digit to see how we extend the process. 9 x 7 4 7

Multiplication I Let’s add another digit to see how we extend the process. 4x7=28 9 x 7 4 7

Multiplication I Let’s add another digit to see how we extend the process. carry the 2 4x7=28 9 x 7 4 7 8 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry the 2 7x7=49, 4x7=28 9 x 7 4 7 8 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry the 2 7x7=49, 49+2=51 4x7=28 9 x 7 4 7 8 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry carry the 5 the 2 7x7=49, 49+2=51 4x7=28 9 7 4 1 7 8 x record the 1 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry carry the 5 the 2 9x7=63, 9 7x7=49, 49+2=51 4x7=28 7 4 1 7 8 x record the 1 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry carry the 5 the 2 9x7=63, 7x7=49, 63+5=68 49+2=51 4x7=28 9 7 4 1 7 8 x record the 1 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry carry the 5 the 2 9x7=63, 7x7=49, 63+5=68 49+2=51 4x7=28 9 6 carry the 6 x 8 7 4 1 7 8 record record the 1 the 8 record the 8

Multiplication I Let’s add another digit to see how we extend the process. carry carry the 5 the 2 9x7=63, 7x7=49, 63+5=68 49+2=51 4x7=28 9 6 carry the 6 x 8 7 4 1 7 8 record record the 1 the 8 record the 8 Your turn: Multiply 8 6 7 x 7 8 6 7 x

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. 9 6 7 4 x 8 6 7 8 1

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. 9 7 4 x 6 7

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. carry the 2 4x7=28 9 7 4 x 6 7 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. carry the 2 7x7=49, 4x7=28 49+2=51 9 7 4 x 6 7 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. carry the 5 carry the 2 7x7=49, 4x7=28 49+2=51 9 7 4 x 6 7 8 1 record the 1 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. carry the 5 carry the 2 9x7=63, 7x7=49, 4x7=28 63+5= 68 49+2=51 9 7 4 x 6 7 8 1 record the 1 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. carry the 5 carry the 2 9x7=63, 7x7=49, 4x7=28 63+5= 68 49+2=51 9 6 carry the 6 7 4 x 8 6 7 8 1 record record the 1 the 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. carry the 5 carry the 2 9x7=63, 7x7=49, 4x7=28 63+5= 68 49+2=51 9 6 carry the 6 7 4 x 8 6 7 8 1 record record the 1 the 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. 6 When this is completed, we proceed with the multiplication to carry the next digit of the bottom number. the 6 Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. carry the 5 carry the 2 9x7=63, 7x7=49, 4x7=28 63+5= 68 49+2=51 9 7 4 x 8 6 7 8 1 record record the 1 the 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. x 6 carry the 6 8 7 4 6 9 7 8 1 record record the 1 the 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. 4x6=24 x 6 carry the 6 8 7 4 6 9 7 8 1 record record the 1 the 8 record the 8

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. carry the 2 4x6=24 x 6 carry the 6 8 7 4 6 9 7 8 1 record record the 1 the 8 4 record the 8 ←record

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. carry the 2 7x6=42, 4x6=24 42+2=44 x 6 carry the 6 8 7 4 6 9 7 8 1 record record the 1 the 8 4 record the 8 ←record

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. carry the 4 carry the 2 7x6=42, 4x6=24 42+2=44 x 6 carry the 6 8 7 4 6 9 7 8 1 record record the 1 the 8 4 4 record the 8 ←record

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. carry the 4 carry the 2 7x6=42, 4x6=24 9x6=54 54+4= 58 42+2=44 x 6 carry the 6 8 7 4 6 9 7 8 1 record record the 1 the 8 4 4 record the 8 ←record

Multiplication I We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number. For example, we start the multiplication as before by multiplying the top with the bottom unit-digit. When this is completed, we proceed with the multiplication to the next digit of the bottom number. Because we are in a 5 place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. carry the 4 carry the 2 7x6=42, 4x6=24 9x6=54 54+4= 58 42+2=44 x 6 carry the 6 8 8 7 4 6 9 7 8 1 record record the 1 the 8 4 4 record the 8 ←record

Multiplication I We treat the multiplication of two carry carry multiple digit numbers as separate the 4 the 2 problems of multiplying with a 7x6=42, 4x6=24 9x6=54 single digit number. 54+4= 58 42+2=44 For example, 7 4 9 we start the multiplication as before by multiplying the top 6 x 7 with the bottom unit-digit. 8 8 1 6 When this is completed, we proceed with the multiplication to carry record record record the next digit of the bottom number. the 6 the 8 the 1 the 8 Because we are in a 8 4 + 5 4 ←record place value system, the result of the multiplication must be placed in the correct slots, so it is shift one place to the left. Finally, we obtain the answer by adding the two columns.

Multiplication I We treat the multiplication of two carry carry multiple digit numbers as separate the 4 the 2 problems of multiplying with a 7x6=42, 4x6=24 9x6=54 single digit number. 54+4= 58 42+2=44 For example, 7 4 9 we start the multiplication as before by multiplying the top 6 x 7 with the bottom unit-digit. 8 8 1 6 When this is completed, we proceed with the multiplication to carry record record record the next digit of the bottom number. the 6 the 8 the 1 the 8 Because we are in a 8 4 + 5 4 ←record place value system, the 6 5 5 8 2 result of the multiplication must be placed in the correct slots, so it is shift one place to the left. Finally, we obtain the answer by adding the two columns.

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end.

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400,

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400, 400 x 60 = 24,000, 400 x 600 = 240,000

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400, 400 x 60 = 24,000, 400 x 600 = 240,000 In other words, we may cut all the trailing 0’s and put them aside, multiply the front parts of the two numbers, then paste the stripped 0’s back for the final answer.

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400, 400 x 60 = 24,000, 400 x 600 = 240,000 In other words, we may cut all the trailing 0’s and put them aside, multiply the front parts of the two numbers, then paste the stripped 0’s back for the final answer. Hence, to multiply 97,400,000 x 6,700

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400, 400 x 60 = 24,000, 400 x 600 = 240,000 In other words, we may cut all the trailing 0’s and put them aside, multiply the front parts of the two numbers, then paste the stripped 0’s back for the final answer. Hence, to multiply cut the trailing 0’s, 97,400,000 x 6,700 put them aside: 0,000,000

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400, 400 x 60 = 24,000, 400 x 600 = 240,000 In other words, we may cut all the trailing 0’s and put them aside, multiply the front parts of the two numbers, then paste the stripped 0’s back for the final answer. Hence, to multiply cut the trailing 0’s, 97,400,000 x 6,700 put them aside: 0,000,000 multiply: 974 x 67 → 65,258,

Multiplication I Finally we note that we may simplify the steps when multiplying numbers with trailing 0’s, i.e. 0’s to the right end. For example, 40 x 6 = 240, 400 x 6 = 2400, 400 x 60 = 24,000, 400 x 600 = 240,000 In other words, we may cut all the trailing 0’s and put them aside, multiply the front parts of the two numbers, then paste the stripped 0’s back for the final answer. Hence, to multiply cut the trailing 0’s, 97,400,000 x 6,700 put them aside: 0,000,000 multiply: 974 x 67 → 65,258, Paste the 0’s back for the final answer so 97,400,000 x 6,700 = 652,580,000,000

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