Post on 25-May-2015
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).