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Product in Math

The product in math is a term that refers to the result of a multiplication problem. In this section, we will learn the term product along with its properties and examples in detail.

What is product in math?

Product in math can be defined as the multiplication of two or more numbers together. In other words, an expression that identifies factors to be multiplied.

The product is the result that we get on multiplying the two numbers multiplier and multiplicand together.

The number which is left of the multiplication sign is called the multiplier, and the number which is right of the multiplication sign is called the multiplicand. Both the multiplier and the multiplicand are also known as factor.

How to Find Product

We get the product of two numbers by applying the mathematical multiplication operation (× or * or .) between two or more numbers. For example:

9×7=63

Here, 63 is the product of 9 and 7.

Similarly,

4×5×8=160

Here, 160 is the product of 4, 5 and 8.

We can also find the product of two numbers by repeated addition method. It means add the number into itself up to multiplier times. This method applies only when we want to find the product of two small numbers.

It means, add b up to a times or vice-versa.

But it is not a traditional method of multiplying. For example, if we want the of 6 and 7, we can add the number 6 up to seven times.

6+6+6+6+6+6+6=42

Or

7+7+7+7+7=42

On multiplying the numbers 6 and 7, we get the same result.

6×7=42

Note: At the last of this section, we have written the steps of how to find the product of two 2-digit numbers and the product of two 3-digit numbers.

Product of Two Integers

Integers includes both positive and negative numbers. The multiplier or multiplicand may hold a positive or a negative sign before the number. No sign before a number represents a positive number. If the number holds a positive or a negative sign, they follow the rules, given in the following table.

The above table represents that:

• The multiplication of two negative numbers gives a positive
• The multiplication of two positive numbers also gives a positive
• The multiplication of a positive and a negative number gives a negative
• The multiplication of a negative and a positive number gives a negative

Let’s see some examples based on the above rules.

Examples

15×5=75

-3×-9=27

-14×5=70

6×-12=72

Product of Two Decimal Numbers

A decimal number is the number that contains a decimal point (.). For example, 23.56 is a decimal number.

We can also find the product of two decimal numbers by using the following steps:

• Ignore the decimal point of both numbers, for a moment.
• Multiply both numbers as an integer. After performing multiplication, we get the product.
• In the multiplier and multiplicand, count the total decimal digits from the right.
• In the product, count the same number of digits from the right, and put a decimal point there.
• Now, we have the product of two decimal numbers.

Example: Find the product of 23.3 and 12.21.

Solution:

In the question, there are two decimal numbers 23.3 and 12.21.

• Ignore the decimal point from both numbers. The numbers become 233 and 1221.
• Multiply both numbers together.
  233×1221=284493
  We get the product 284493.
• In the multiplier and multiplicand, count the total decimal digits from the right. There is one digit after the decimal in the multiplier, and two digits after decimal in the multiplicand. So, there are a total of three (1 + 2) decimal digits.
• In the product (284493), count the same number of decimal digits (three) from the right, and put the decimal point there. We get 493 as a product of 23.3 and 12.21.

Product of Two Fractions

Fractions are the numbers that are in the form of numerator and denominator.

To find the product of two fractions, multiply the numerator of the multiplier by the numerator of the multiplicand, and the denominator of the multiplier by the denominator of the multiplicand. In the terms of formula, we can write the above statement as:

The product of a fractional number and its multiplicative inverse is always 1. Suppose, a fraction number is and its multiplicative inverse is then:

Always consider 1 if there is no denominator in the fraction. Digit 3 is same as in the fractional form.

Sometimes, we need to simplify the fraction if the fraction is divisible by the denominator or the numerator of the multiplier is divisible by the denominator of the multiplicand or vice-versa.

Note: The numerator and denominator must be divisible by the same number.

Product of Two Complex Numbers

A complex number is a number that can be expressed in the form of (a+bi) or (a-bi), where a and b are real numbers and i is an imaginary number.

We can find the product of two complex number by using the distributive property. Remember the following points about i.

• i⁰=1
• i¹=i
• i²=-1 or i=√-1
• i³=-i
• i⁴=1
• i⁵=i

In general terms, the product of two complex numbers (a+bi) and (c+di) is:

Let’s see an example.

Example: Find the product of two complex numbers (3-2i) and (-1+4i).

Solution:

(3-2i)×(-1+4i)=3×(-1)+3×(4i)-(2i)×(-1)-(2i)×(4i)
=-3+12i+2i-8i²
=-3+14i-8i²

Putting the value of i²=-1, we get:

-3+14i-8×(-1)
-3+14i+8
5+14i

The product of two complex numbers (3-2i) and (-1+4i) is (5+14i).

Product Properties of Square Root

• √a×√b=√a×b
• √a×√a=√a²=a
• a×√b=a√b
• √a×b=b√a
• a√x×b√y=a×b√(x×y)

Properties of Product

There are four basic properties of the product:

• Associative Property
• Commutative Property
• Identity Property
• Distributive Property

Associative Property

When we multiply three or more numbers together, the product is the same regardless of which two are multiplied first.

Commutative Property

The order of multiplication does not affect the product.

Identity Property

We get the same number if a number is multiplied by 1. So, 1 is called the multiplicative identity.

Distributive Property

The property is known as the distributive property of multiplication over addition. It states that the sum multiplied by a number, we can multiply each piece of the sum by the number first and then add the results.

Some other properties of the product are:

• The product of the two same numbers is called the square of the number.
• We get 0 as a product if a number is multiplied by 0.
• If the multiplier and multiplicand are 1, the product will also be 1.
• Products are not unique.

Product of Two Large Numbers

Once we catch the logic, then it is very easy to remember. For better understanding, we have written the steps of the multiplication of two large numbers.

Note: The following methods apply when the multiplier and the multiplicand have the same number of digits.

Product of Two Single-Digit Numbers

To find the product of two single-digit numbers, we must remember the tables or multiplication chart up to 10. It makes it easy to find the product of two numbers. Even we can find the product without using the pen and copy. The following chart shows the tables from 1 to 10.

Product of Two 2-Digit Numbers

It includes three steps. Here, we have taken four alphabets (a, b, c, d) as a number for easy understanding.

Suppose, we want the product of ab×cd, then we must follow the steps given below:

Let’s implement the above steps in an example.

Example: What will be the product of 47×56?

Step 1: (b×d)

7×6=42

Write 2 in the answer and take 4 as carry over to next step.

(Updated Answer: 2)

Step 2: [(a×d)+(b×c)]+carry over of the previous step if any

[(4×6)+(5×7)]+4=63

Write 3 in the answer and take 6 as carry over to next step.

(Updated Answer: 32)

Step 3: (a×d)+carry over of the previous step if any

(4×5)+6=26

Write 26 in the answer.

(Updated Answer: 2632)

The product of (47×56) is 2632.

Note: Always write answer right to left.

Product of Two 3-Digit Numbers

It includes five steps. Here, we have taken six alphabets (a, b, c, d, e, f) as a number for easy understanding.

Suppose, we want the product of abc×def, then we must follow the steps given below:

Let’s implement the above steps in an example.

Example: What will be the product of 624×315?

Step 1: (c×f)

4×5=20

Write 0 in the answer and take 2 as carry over to next step.

(Updated Answer: 0)

Step 2: [(b×f)+(c×e)]+carry over of the previous step if any

[(2×5)+(4×1)]+2=16

Write 6 in the answer and take 1 as carry over to next step.

(Updated Answer: 60)

Step 3: [(a×f)+(b×e)+(c×d)]+carry over of the previous step if any

[(6×5)+(2×1)+(4×3)]+1=45

Write 5 in the answer and take 4 as carry over to next step.

(Updated Answer: 560)

Step 4: [(a×e)+(b×d)]+carry over of the previous step if any

[(6×1)+(2×3)]+4=16

Write 6 in the answer and take 1 as carry over to next step.

(Updated Answer: 6560)

Step 5: (a×d)+carry over of the previous step if any

(6×1)+1=19

Write 19 in the answer.

(Updated Answer: 19560)

The product of (624×315) is 196560.

Similarly, we can also find the product of two 4-digit numbers by using the following steps.

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