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Identity Property of Multiplication

The Identity Property of Multiplication is the rule that any number multiplied by 1 equals that same number. In other words, multiplying by 1 doesn't change the value.

The Identity Property of Multiplication states that for any real number aa, the equation a×1=aa \times 1 = a holds true. The number 1 is called the multiplicative identity because it preserves the value of any number it is multiplied by. This property applies to whole numbers, fractions, decimals, negative numbers, and all other real numbers.

Key Formula

a×1=1×a=aa \times 1 = 1 \times a = a
Where:
  • aa = any real number
  • 11 = the multiplicative identity

Worked Example

Problem: Show that the Identity Property of Multiplication holds for the number −7.5.
Step 1: Write the property using the given number.
−7.5×1=  ?-7.5 \times 1 = \;?
Step 2: Multiply −7.5 by 1. Since multiplying by 1 leaves the number unchanged, the result is −7.5.
−7.5×1=−7.5-7.5 \times 1 = -7.5
Step 3: Check the reverse order to confirm the commutative case as well.
1×(−7.5)=−7.51 \times (-7.5) = -7.5
Answer: Both −7.5×1-7.5 \times 1 and 1×(−7.5)1 \times (-7.5) equal −7.5-7.5, confirming the Identity Property of Multiplication.

Why It Matters

This property comes up constantly when you simplify expressions in algebra. For instance, when you see a variable with no coefficient written in front of it, like xx, that really means 1×x1 \times x. Understanding the multiplicative identity also helps when working with fractions — multiplying by 33\frac{3}{3} is the same as multiplying by 1, which is how you create equivalent fractions without changing a value.

Common Mistakes

Mistake: Confusing the identity property of multiplication with the identity property of addition.
Correction: The multiplicative identity is 1 (multiplying by 1 keeps the number the same), while the additive identity is 0 (adding 0 keeps the number the same). They are different operations with different identity elements.
Mistake: Thinking that multiplying by 0 is the same as multiplying by 1.
Correction: Multiplying by 0 always gives 0 (the zero property), not the original number. Only multiplying by 1 leaves the number unchanged.

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