Multiplication II

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In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

Properties of Multiplication

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

Properties of MultiplicationWe note from before before that

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

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .

Properties of MultiplicationWe note from before before that

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

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .

Properties of MultiplicationWe note from before before that

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

For example, A*B*0*C = 0 where A, B ,and C are numbers.

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .

Properties of MultiplicationWe note from before before that

* (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 example, A*B*0*C = 0 where A, B ,and C are numbers.

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .

Properties of MultiplicationWe note from before before that

* (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.In other words, 1 is the “preserver” in multiplication, It preserves anything multiplies with it .

For example, A*B*0*C = 0 where A, B ,and C are numbers.

In the last section we covered the arithmetic mechanics of multiplying two numbers in the vertical format where in modern day most of this work is delegated to calculators or software.

Multiplication II

In mathematics, we are also interested in the properties and relations.

In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .

Properties of MultiplicationWe note from before before that

For example, A*1*B*1*C = A*B*C.

* (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.In other words, 1 is the “preserver” in multiplication, It preserves anything multiplies with it .

For example, A*B*0*C = 0 where A, B ,and C are numbers.

Multiplication II

3 copies = 2 copies

* We noted that

Multiplication II

3 copies = 2 copies

so that 3 x 2 = 2 x 3

* We noted that

and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication II

3 copies = 2 copies

so that 3 x 2 = 2 x 3

* We noted that

* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).

and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication II

3 copies = 2 copies

so that 3 x 2 = 2 x 3

* We noted that

* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).

For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.

6 12

and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication II

3 copies = 2 copies

so that 3 x 2 = 2 x 3

* We noted that

* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).

Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.

For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.

6 12

and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication II

3 copies = 2 copies

so that 3 x 2 = 2 x 3

* We noted that

* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).

Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.

For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.

6 12

Above observations provide us with short cuts for lengthy multiplication that involves many numbers.

and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication II

3 copies = 2 copies

so that 3 x 2 = 2 x 3

* We noted that

* Similarly, we may easily verify, just as addition, that multiplication is associative, i.e. (A x B) x C = A x (B x C).

Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.

For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.

6 12

Above observations provide us with short cuts for lengthy multiplication that involves many numbers. They also provide ways to double check our answers as shown below.

and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example, 2 x 4 x 3 x 5 = (2 x 4) x (3 x 5)

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example, 2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25

2 x 4 x 3 x 5 x 25

I.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25

2 x 4 x 3 x 5 x 25

10 100I.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

= 3 x 10 x 100= 3,000

Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25

2 x 4 x 3 x 5 x 25

10 100I.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

= 3 x 10 x 100= 3,000

Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25

2 x 4 x 3 x 5 x 25

10 100

2 x 4 x 3 x 5 x 25

50

20

I. II.

Multiplication IIi. For a lengthy multiplication, multiply in pairs.

For example,

10

2 x 4 x 3 x 5 = (2 x 4) x (3 x 5) = 8 x 15 = 120

or 2 x 3 x 4 x 5

12

= 10 x 12 = 120

ii. Look for “even numbers” x “multiples of 5”, these produce “multiples of 10” with trailing 0’s.

Example A. a. Multiply 2 x 4 x 1 x 3 x 5 x 1 x 25. Do it in two different orders to confirm the answer.

= 3 x 10 x 100= 3,000

Drop the 1’s: 2 x 4 x 1 x 3 x 5 x 1 x 25 = 2 x 4 x 3 x 5 x 25

2 x 4 x 3 x 5 x 25

10 100

= 20 x 3 x 50= 3,000

2 x 4 x 3 x 5 x 25

50

20

I. II.

Multiplication IIEven if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

= 136 x 2

= 272

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

= 136 x 2

= 272

Doing it in pairs:(3 x 3) x (2 x 7) x 2

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

= 136 x 2

= 272

Doing it in pairs:(3 x 3) x (2 x 7) x 2

= 9 x 14 x 2

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

= 136 x 2

= 272

Doing it in pairs:(3 x 3) x (2 x 7) x 2

= 9 x 14 x 2

= 272

= 9 x 28

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

Multiplication II

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

= 136 x 2

= 272

Doing it in pairs:(3 x 3) x (2 x 7) x 2

= 9 x 14 x 2

= 272

= 9 x 28

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

2 + 2 + 2 = 3 x 2

3 copies

We simplify the notation forrepetitive additions as:

Multiplication II

We simplify the notation forrepetitive multiplication as:

= 9 x 2 x 7 x 2

b. Multiply 3 x 3 x 2 x 7 x 2. Do it in the order that’s given first, then do it in pairs, to confirm the answer.

3 x 3 x 2 x 7 x 2Doing it in the order that’s given:

= 18 x 7 x 2

= 136 x 2

= 272

Doing it in pairs:(3 x 3) x (2 x 7) x 2

= 9 x 14 x 2

= 272

= 9 x 28

Even if there is no “easy picking” as in the previous example, it’s shorter to multiply in pairs then multiply one-by-one in order.

2 + 2 + 2 = 3 x 2

3 copies

We simplify the notation forrepetitive additions as:

2 * 2 * 2 = 23 = 8

3 copies

Multiplication IIAbout the Notation

Multiplication II

In the notation

= 2 * 2 * 223 = 8

About the Notation

Multiplication II

In the notation

= 2 * 2 * 223this is the base = 8

About the Notation

Multiplication II

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

= 8

About the Notation

Multiplication II

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

About the Notation

Multiplication II

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

About the Notation

Recall that for repetitive addition, we write

3 copies

2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) with 3 to one the side.

Multiplication II

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

About the Notation

Recall that for repetitive addition, we write

3 copies

So for repetitive multiplication, to distinguish it from addition, we store the number of repetition in the upper corner.

2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) with 3 to one the side.

Multiplication II

In the notation

= 2 * 2 * 223this is the base

this is the exponent, or the power, which is the number of repetitions.

We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

= 8

About the Notation

Recall that for repetitive addition, we write

3 copies

3 copies

So for repetitive multiplication, to distinguish it from addition, we store the number of repetition in the upper corner.

2 + 2 + 2 as 3 x 2 = 3(2) = 2(3) with 3 to one the side.

Hence, we write 2 * 2 * 2 as 23.

Multiplication IIExample B. Calculate the following.

a. 3(4) b. 34 c. 43

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12

a. 3(4) b. 34 c. 43

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3a. 3(4) b. 34 c. 43

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

a. 3(4) b. 34 c. 43

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

a. 3(4) b. 34 c. 43

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 12

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3 = 2*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 12

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3 = 2*3*3

= 6*3

= 18

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 12

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3 = 2*3*3

= 6*3

= 18

= 2*2*3*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 12

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3 = 2*3*3

= 6*3

= 18

= 2*2*3*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 4 * 9= 12 * 3

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3 = 2*3*3

= 6*3

= 18

= 2*2*3*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 36 * 3

= 4 * 9= 12 * 3

= 108

Multiplication IIExample B. Calculate the following.

= 12 = 3*3*3*3

= 9 9*

= 81

= 4 * 4 * 4a. 3(4) b. 34 c. 43

= 16 * 4= 64

= 2*2*3 = 2*3*3

= 6*3

= 18

= 2*2*3*3*3

d. 22 x 3 e. 2 x 32 f. 22 x 33

= 36 * 3

= 4 * 9= 12 * 3

= 108

Problems d, c and e are the same as 22(3), 2(32), and 22(33).