What's the Product? Trivia Quiz

Answer: Pi

The lower case pi denotes the ratio of the circumference to the diameter of a circle. It is an irrational number, 3.142... On the contrary, the upper case pi denotes the product of a series of numbers.

Answer: The product of an even number and an odd number is an odd number.

Let 2n be the even number and 2m + 1 be the odd number. (2n)(2m+1) = 4mn+2n. 4mn is even since it is divisible by 4, which means it is also divisible by 2. 2n is also even. Therefore, we can conclude that 4mn+2n can also be divided completely by 2. Hence, it is an even number.

Answer: 10!

10! means 10 factorial. It is equivalent to 10 x 9 x ... x 2 x 1. In general, n! = n x (n-1) x (n-2) x ... x 2 x 1.

Answer: 1/n

Take any number and multiply it with 1, we will get back the number itself. Let the number be n. n x 1 = n. So, 1 is the multiplicative identity.

To find the multiplicative inverse, we need to ask ourselves, what number should we multiply with n so that the product is 1? Let the multiplicative inverse be m. n x m = 1. Solving for m, m = 1/n.

Answer: Commutative

The commutative property states that a x b = b x a.

The associative property states that (a x b) x c = a x (b x c).

The distributive property is illustrated by the equation a(b+c) = ab + ac.

The identity property is a x 1 = 1 x a = a.

Answer: 2^11

2^5 x 2^6 = (2 x 2 x 2 x 2 x 2) x (2 x 2 x 2 x 2 x 2 x 2) = 2^11

Here, 2 is multiplied 11 times.

On the other hand, (2^5)^6 = 2^30.

Answer: 13

(2+3i)(2-3i) = 4 -6i + 6i -9i^2 = 4 -9i^2 = 4 - 9(-1) = 4 + 9 = 13.

Since i is the square root of -1, i^2 = -1.

Answer: 2 rows and 4 columns

Generally, AB is not equal to BA. Unlike multiplication of real numbers, matrix multiplication is not commutative.

Answer: No

Given two vectors, a = (a1, a2, a3) and b = (b1, b2, b3)

The cross product of vectors a and b = a x b = (a1, a2, a3) x (b1, b2, b3) = (a2xb3-a3xb2, a3xb1-a1xb3, a1xb2-a2xb1). Notice that this product is in the form of vectors. The alternative name of cross products is vector products.

On the other hand, the dot product of vectors a and b = a.b = a1xb1 + a2xb2 + a3xb3. Observe that this product is in the form of scalar values. The dot product is also known as the scalar product.

Answer: 41 x 32

The goal here is to form two 2-digit numbers so that their product is as big as possible.

Clearly, we should have the numbers in the form of forty-something times thirty something to maximize our product. Now the problem is should it be 41 x 32 or 42 x 31? By direct calculation, 41 x 32 = 1312 is greater than 42 x 31 = 1302. So the correct answer is 41 x 32.

We can also solve this problem by using algebra. Let's say the 4 numbers are a, b, c and d. The two 2-digit numbers are 10a+b and 10c+d. We want to maximize (10a+b)(10c+d). We expand the two terms to get 100ac + 10ad + 10bc + bd.

The coefficient of ac is 100, which means we should assign greater numbers to both a and b. So, a = 4 and c = 3 will do.

Substituting a = 4 and c = 3 into 100ac + 10ad + 10bc + bd, we arrive at 1200 + 40d + 30b + bd. Ignoring 1200, now we only need to concentrate on the remaining 3 terms, namely 40d + 30b + bd. Since we have used 4 and 3, this leaves us with the numbers 1 and 2.

So, our next goal is to determine whether we should assign b = 1, d = 2 or b = 2, d = 1.

No matter which pair we choose, the last term, bd will always be 1 x 2 or 2 x 1 = 2.

So again, ignoring the term bd, this leaves us with the remaining 2 terms, namely 40d and 30b. It is obvious that if d = 2 and b = 1, then 40d + 30b will yield a greater value instead of choosing d = 1 and b = 2.

Now, we have a = 4, c = 3, d = 2 and b = 1.

(10a+b)(10c+d)
= 100ac + 10ad + 10bc + bd

41 x 32 = 1312

To confirm our answer, 42 x 31 = 1302, which is clearly less than the product of 41 and 32, which is 1312.