1.6 multiplication i w

Multiplication I

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

We simplify the notation for adding the same quantity repeatedly.

Multiplication I

We simplify the notation for adding the same quantity repeatedly.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

Multiplication I

We simplify the notation for adding the same quantity repeatedly.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

Multiplication I

We simplify the notation for adding the same quantity repeatedly.

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.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

Multiplication I

We simplify the notation for adding the same quantity repeatedly.

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.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

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.

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.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

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.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

the multiplicands 2 and 3are called factors (of 6).

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

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.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

the multiplicands 2 and 3are called factors (of 6).

the result 6 is called the product(of 2 and 3).

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

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.

We call this operation multiplication and we say that “3 times 2 is 6” or “3 multiplied with 2 is 6”.

the multiplicands 2 and 3are called factors (of 6).

the result 6 is called the product(of 2 and 3).

Note that 3 copies = 2 copies

so that 3 x 2 = 2 x 3.

2 + 2 + 2 = 6

3 copies

as 3 x 2 or 3*2 or 3(2) = 6.

For example, we shall write

In the expression: 3 x 2 = 2 x 3 = 6

(Note: 1 and 6 are also factors of 6 because 1 x 6 = 6 x 1 = 6.)

In general, just as addition, multiplication is commutative, i.e. A x B = B x A.

Multiplication I

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

Multiplication I

* (0 * x = 0 * x = 0) The product of zero with any number is 0.

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

Multiplication I

* (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.

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

Multiplication I

* For the products with 9 as a factor, the sum of their digits is 9.

* (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.

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

Multiplication I

* For the products with 9 as a factor, the sum of their digits is 9.

6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81

For example,

* (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.

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

all have digit sum equal to 9,

Multiplication I

* For the products with 9 as a factor, the sum of their digits is 9.

6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81

For example,

i.e. 5 + 4 = 9,

* (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.

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

all have digit sum equal to 9,

Multiplication I

* For the products with 9 as a factor, the sum of their digits is 9.

6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 9 x 9 = 81

For example,

i.e. 5 + 4 = 9, 6 + 3 = 9

* (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.

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

7 + 2 = 9, 8 + 1 = 9all have digit sum equal to 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

6636 6742 6848 69547749 7856 7963

8864 8972 9981

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

Multiplication I

6636 6742 6848 69547749 7856 7963

8864 8972 9981

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

6 x 7 = 42 (= 7 x 6)

For example,

7 x 8 = 56 (= 8 x 7).

Multiplication I

6636 6742 6848 69547749 7856 7963

8864 8972 9981

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

6 x 7 = 42 (= 7 x 6)

For example,

The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etcare called even numbers.

7 x 8 = 56 (= 8 x 7).

Multiplication I

6636 6742 6848 69547749 7856 7963

8864 8972 9981

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

6 x 7 = 42 (= 7 x 6)

For example,

The numbers with 2 as a factor: 0, 2, 4, 6, 8,…etcare 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.

7 x 8 = 56 (= 8 x 7).

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

47

7x

For example,

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

47

i. Starting from the right, multiply the two unit-digits,

7x

For example,

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.

record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

7x

For example,

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.

i. 4x7=28 record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

7x8

record the 8,

carry the 2

For example,

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.

i. 4x7=28 record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

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.

i. 4x7=28 record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

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.

i. 4x7=28 ii. 7x7=49,record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

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.

i. 4x7=28 ii. 7x7=49, 49+2=51

add the previous carry to the product,

record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

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.

i. 4x7=28 ii. 7x7=49,

1

record the 1,

5

carry the 5

49+2=51

add the previous carry to the product,record the unit-digit of this sum and carry the 10’s digit of this sum.

record the unit-digit of the product, and carry the 10’s digit of the product.

Multiplication I

The Vertical Format

47

i. Starting from the right, multiply the two unit-digits,

ii. Multiply the next digit of the double digit number to the single digit,

7x8

record the 8,

carry the 2

For example,

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.

i. 4x7=28 ii. 7x7=49,

1

record the 1,

5

carry the 5

49+2=51

add the previous carry to the product,record the unit-digit of this sum and carry the 10’s digit of this sum.

record the unit-digit of the product, and carry the 10’s digit of the product.

To multiply a longer number against a single digit number, repeat step ii until all the digits are multiplied.

Multiplication I

47

7x

9

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

4x7=28

9

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

carry the 2

4x7=28

9

Let’s add another digit to see how we extend the process.

Multiplication I

8

record the 8

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

9

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

8

carry the 2

4x7=28 7x7=49,

49+2=51

9

Let’s add another digit to see how we extend the process.

Multiplication I

record the 8

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63,

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5=68

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5=68

8

record the 8

carry the 6

6

Let’s add another digit to see how we extend the process.

Multiplication I

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5=68

8

record the 8

carry the 6

6

Let’s add another digit to see how we extend the process.

Your turn: Multiply 8 6

7x8 6

7x7

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

47

7x

81

9

866

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

9

6

For example,

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28

9

6

For example,

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49, 49+2=51

9

6

For example,

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

49+2=51

9

6

For example,

Multiplication Icarry the 5

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

6

For example,

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

8

record the 8

carry the 6

66

For example,

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

8

record the 8

carry the 6

66

When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

we start the multiplication as before by multiplying the top with the bottom unit-digit.

47

7x

8

record the 8

carry the 2

4x7=28 7x7=49,

1

record the 1

carry the 5

49+2=51

9

9x7=63, 63+5= 68

8

record the 8

carry the 6

66

When this is completed, we proceed with the multiplication to the next digit of the bottom number.

For example,

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.

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

1

record the 1

9

8

record the 8

carry the 6

6

6

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

4x6=24

1

record the 1

9

8

record the 8

carry the 6

6

6

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

4x6=24

1

record the 1

←record

9

8

record the 8

carry the 6

6

6

carry the 2

4

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

4x6=24 7x6=42,

1

record the 1

←record

42+2=44

9

8

record the 8

carry the 6

6

6

carry the 2

4

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44

9

8

record the 8

carry the 6

6

6

carry the 2

44

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44

9

9x6=54 54+4= 58

8

record the 8

carry the 6

6

6

carry the 2

44

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44

9

9x6=54 54+4= 58

8

record the 8

carry the 6

6

6

carry the 2

4485

Multiplication I

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44

9

9x6=54 54+4= 58

8

record the 8

carry the 6

6

6

carry the 2

Finally, we obtain the answer by adding the two columns.

4485

Multiplication I

+

We treat the multiplication of two multiple digit numbers as separate problems of multiplying with a single digit number.

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.

For example,

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.

47

7x8

record the 8

carry the 4

4x6=24 7x6=42,

1

record the 1

←record

42+2=44

9

9x6=54 54+4= 58

8

record the 8

carry the 6

6

6

carry the 2

Finally, we obtain the answer by adding the two columns.

44

85

85 2 6 5

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.

40 x 6 = 240,

Multiplication I

For example,

400 x 6 = 2400,

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

40 x 6 = 240,

Multiplication I

For example,

400 x 6 = 2400,

400 x 60 = 24,000, 400 x 600 = 240,000

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

40 x 6 = 240,

Multiplication I

For example,

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.

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

40 x 6 = 240,

Multiplication I

For example,

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

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

40 x 6 = 240,

Multiplication I

For example,

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 6,700put them aside: 0,000,000

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

40 x 6 = 240,

Multiplication I

For example,

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 6,700put them aside:

multiply: 974 x 67 → 65,258,

0,000,000

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

40 x 6 = 240,

Multiplication I

For example,

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 6,700put them aside:

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

0,000,000