Information about 1.7 multiplication ii w

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

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

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. In mathematics, we are also interested in the properties and relations. Properties of Multiplication We note from before before that * (0 * x = 0 * x = 0) The product of zero with any number is 0.

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. In mathematics, we are also interested in the properties and relations. Properties of Multiplication We note from before before that * (0 * x = 0 * x = 0) The product of zero with any number is 0. In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it .

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. In mathematics, we are also interested in the properties and relations. Properties of Multiplication We note from before before that * (0 * x = 0 * x = 0) The product of zero with any number is 0. In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it . For example, A*B*0*C = 0 where A, B ,and C are numbers.

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. In mathematics, we are also interested in the properties and relations. Properties of Multiplication We note from before before that * (0 * x = 0 * x = 0) The product of zero with any number is 0. In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it . For example, A*B*0*C = 0 where A, B ,and C are numbers. * (1 * x = x * 1 = x) The product of 1 with any number x is x.

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. In mathematics, we are also interested in the properties and relations. Properties of Multiplication We note from before before that * (0 * x = 0 * x = 0) The product of zero with any number is 0. In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it . For example, A*B*0*C = 0 where A, B ,and C are numbers. * (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 .

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. In mathematics, we are also interested in the properties and relations. Properties of Multiplication We note from before before that * (0 * x = 0 * x = 0) The product of zero with any number is 0. In other words, 0 is the “annihilator” in multiplication. It demolishes anything multiplied with it . For example, A*B*0*C = 0 where A, B ,and C are numbers. * (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*1*B*1*C = A*B*C.

Multiplication II * We noted that 3 copies = 2 copies

Multiplication II * We noted that 3 copies = 2 copies so that 3 x 2 = 2 x 3 and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A.

Multiplication II * We noted that 3 copies = 2 copies so that 3 x 2 = 2 x 3 and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A. * 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 II * We noted that 3 copies = 2 copies so that 3 x 2 = 2 x 3 and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A. * 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

Multiplication II * We noted that 3 copies = 2 copies so that 3 x 2 = 2 x 3 and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A. * 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 Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish.

Multiplication II * We noted that 3 copies = 2 copies so that 3 x 2 = 2 x 3 and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A. * 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 Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish. Above observations provide us with short cuts for lengthy multiplication that involves many numbers.

Multiplication II * We noted that 3 copies = 2 copies so that 3 x 2 = 2 x 3 and that in general, just as addition, that multiplication is commutative, i.e. A x B = B x A. * 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 Multiplication being commutative and associative allows us to multiply a long strings of multiplication in any order we wish. 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.

Multiplication II i. For a lengthy multiplication, multiply in pairs.

Multiplication II i. For a lengthy multiplication, multiply in pairs. For example, 2 x 4 x 3 x 5 = (2 x 4) x (3 x 5)

Multiplication II i. 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 II i. 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 12 or 2x3x4x5 10

Multiplication II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10

Multiplication II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 I. 2 x 4 x 3 x 5 x 25

Multiplication II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 10 100 I. 2 x 4 x 3 x 5 x 25

Multiplication II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 10 100 I. 2 x 4 x 3 x 5 x 25 = 3 x 10 x 100 = 3,000

Multiplication II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 10 100 II. I. 50 2 x 4 x 3 x 5 x 25 = 3 x 10 x 100 = 3,000 2 x 4 x 3 x 5 x 25 20

Multiplication II i. 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 12 or 2 x 3 x 4 x 5 = 10 x 12 = 120 10 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 10 100 II. I. 50 2 x 4 x 3 x 5 x 25 = 3 x 10 x 100 = 3,000 2 x 4 x 3 x 5 x 25 = 20 x 3 x 50 = 3,000 20

Multiplication II 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 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. 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.

Multiplication II 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. 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. Doing it in the order that’s given: 3x3x2x7x2

Multiplication II 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. 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. Doing it in the order that’s given: 3x3x2x7x2 =9x2x7x2

Multiplication II 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. 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. Doing it in the order that’s given: 3x3x2x7x2 =9x2x7x2 = 18 x 7 x 2

Multiplication II 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. 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. Doing it in the order that’s given: 3x3x2x7x2 =9x2x7x2 = 18 x 7 x 2 = 136 x 2 = 272

Multiplication II 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. 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. Doing it in the order that’s given: Doing it in pairs: 3x3x2x7x2 (3 x 3) x (2 x 7) x 2 =9x2x7x2 = 18 x 7 x 2 = 136 x 2 = 272

Multiplication II 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. 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. Doing it in the order that’s given: Doing it in pairs: 3x3x2x7x2 (3 x 3) x (2 x 7) x 2 =9x2x7x2 = 9 x 14 x 2 = 18 x 7 x 2 = 136 x 2 = 272

Multiplication II 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. 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. Doing it in the order that’s given: Doing it in pairs: 3x3x2x7x2 (3 x 3) x (2 x 7) x 2 =9x2x7x2 = 9 x 14 x 2 = 18 x 7 x 2 = 136 x 2 = 272 = 9 x 28 = 272

Multiplication II 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. 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. Doing it in the order that’s given: Doing it in pairs: 3x3x2x7x2 (3 x 3) x (2 x 7) x 2 =9x2x7x2 = 9 x 14 x 2 = 18 x 7 x 2 = 136 x 2 = 272 We simplify the notation for repetitive additions as: 3 copies 2+2+2=3x2 = 9 x 28 = 272

Multiplication II 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. 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. Doing it in the order that’s given: Doing it in pairs: 3x3x2x7x2 (3 x 3) x (2 x 7) x 2 =9x2x7x2 = 9 x 14 x 2 = 18 x 7 x 2 = 136 x 2 = 9 x 28 = 272 = 272 We simplify the notation for repetitive additions as: We simplify the notation for repetitive multiplication as: 3 copies 3 copies 2+2+2=3x2 2 * 2 * 2 = 23 = 8

Multiplication II About the Notation

Multiplication II About the Notation In the notation 23 =2*2*2 =8

Multiplication II About the Notation In the notation this is the base 23 =2*2*2 =8

Multiplication II About the Notation In the notation this is the base this is the exponent, or the power, which is the number of repetitions. 23 =2*2*2 =8

Multiplication II About the Notation In the notation this is the base this is the exponent, or the power, which is the number of repetitions. 23 =2*2*2 =8 We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.”

Multiplication II About the Notation In the notation this is the base this is the exponent, or the power, which is the number of repetitions. 23 =2*2*2 =8 We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.” 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 About the Notation In the notation this is the base this is the exponent, or the power, which is the number of repetitions. 23 =2*2*2 =8 We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.” 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. So for repetitive multiplication, to distinguish it from addition, we store the number of repetition in the upper corner.

Multiplication II About the Notation In the notation this is the base this is the exponent, or the power, which is the number of repetitions. 23 =2*2*2 =8 We say that “2 to the power 3 is 8” or that “2 to the 3rd power is 8.” 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. So for repetitive multiplication, to distinguish it from addition, we store the number of repetition in the upper corner. Hence, we write 2 * 2 * 2 as 23. 3 copies

Multiplication II Example B. Calculate the following. a. 3(4) d. 22 x 3 b. 34 e. 2 x 32 c. 43 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 d. 22 x 3 b. 34 e. 2 x 32 c. 43 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 d. 22 x 3 b. 34 = 3*3*3*3 e. 2 x 32 c. 43 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 c. 43 = 9 * 9 d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 c. 43 = 9 * 9 = 81 d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 c. 43 =4*4*4 = 9 * 9 = 81 d. 22 x 3 e. 2 x 32 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 e. 2 x 32 c. 43 =4*4*4 = 16 * 4 = 64 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 e. 2 x 32 c. 43 =4*4*4 = 16 * 4 = 64 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 e. 2 x 32 = 2*3*3 c. 43 =4*4*4 = 16 * 4 = 64 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 e. 2 x 32 = 2*3*3 = 6*3 = 18 c. 43 =4*4*4 = 16 * 4 = 64 f. 22 x 33

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 e. 2 x 32 = 2*3*3 = 6*3 = 18 c. 43 =4*4*4 = 16 * 4 = 64 f. 22 x 33 = 2*2*3*3*3

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 c. 43 =4*4*4 = 16 * 4 = 64 e. 2 x 32 = 2*3*3 f. 22 x 33 = 2*2*3*3*3 = 6*3 = 18 = 4 *9 *3

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 c. 43 =4*4*4 = 16 * 4 = 64 e. 2 x 32 = 2*3*3 f. 22 x 33 = 2*2*3*3*3 = 6*3 = 18 = 4 *9 *3 = 36 * 3 = 108

Multiplication II Example B. Calculate the following. a. 3(4) = 12 b. 34 = 3*3*3*3 = 9 * 9 = 81 d. 22 x 3 = 2*2*3 = 12 c. 43 =4*4*4 = 16 * 4 = 64 e. 2 x 32 = 2*3*3 f. 22 x 33 = 2*2*3*3*3 = 6*3 = 18 = 4 *9 *3 = 36 * 3 = 108 Problems d, c and e are the same as 22(3), 2(32), and 22(33).

1.7 Linear morphisms II ... Lemma 1.7.2. Let f : V !W be a ... is the composite morphism and on the RHS of the equation we are considering multiplication of

Read more

... Section 1.7 - Multiplication & Division of Real Numbers - Page 3 ... (w + 2) = 4w 10; 9 ... Section 1.7 - Multiplication & Division of Real Numbers ...

Read more

Convention 1.7. Henceforth, except ... ii) Linear transformations f: V=U!W. Proof. ... With addition and scalar multiplication de ned as in 2.2, Hom(V;W)

Read more

−1− √ −7; one of the roots for instance is A+B: it may not look like a real number, but it turns out ... Multiplication (a+bi)(c+di) = (ac−bd ...

Read more

Division Facts II Dividing with 2, 5, and 9 Special Quotients Dividing ... Multiplication can help you divide by 3, 4, 5, 6, 7, and 8. What’s 28 ÷ 7?

Read more

... 7 ∙ 12 ∙ 11 = 924 Gesamtdiff.: 10 ( 5 + 4 + 1 ) 7 ∙ 13 ∙ 10 = 910 Gesamtdiff.: 12 ( 6 + 3 + 3 ) ... 0 ∙ 29 ∙ 1 = 0 Gesamtdiff.: ...

Read more

FAST MULTIPLICATION: ALGORITHMS AND IMPLEMENTATION ... Gary W. Bewick ... 1.7 Reduction of 8 partial products with 4-2 counters.::: :: :: :: :: :: 10

Read more

II.1 Algebren 2.1.1 Deﬁnition ... Das w¨are ein Funktor, ... 2.1.7 Beispiel a) Es sei K ein K¨orper. Der Ring A:= {a b 0 a | a,b∈ K} ⊆ M 2(K) ist eine

Read more

## Add a comment