625 in Binary

The process of converting 625 from decimal to binary involves dividing the number 625 by 2. Here, it is divided by 2 because the binary number system uses only 2 digits (0 and 1). The quotient becomes the dividend in the next step, and the process continues until the quotient becomes 0.

This is a commonly used method to convert 625 to binary. In the last step, the remainder is noted down bottom side up, and that becomes the converted value.

For example, the remainders noted down after dividing 625 by 2 until getting 0 as the quotient form the binary number 1001110001. Remember, the remainders here have been written upside down.

In the table shown below, the first column shows the binary digits (1 and 0) as 1001110001.

The second column represents the place values of each digit, and the third column is the value calculation, where the binary digits are multiplied by their corresponding place values.

The results of the third column can be added to cross-check if 1001110001 in binary is indeed 625 in the decimal number system.

625 can be converted easily from decimal to binary. The methods mentioned below will help us convert the number. Let’s see how it is done.

Expansion Method: Let us see the step-by-step process of converting 625 using the expansion method.

Step 1 - Figure out the place values: In the binary system, each place value is a power of 2. Therefore, in the first step, we will ascertain the powers of 2.

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

2⁹ = 512

Since 512 is the largest power of 2 less than 625, we stop at 2^9 = 512.

Step 2 - Identify the largest power of 2: In the previous step, we stopped at 2^9 = 512. This is because in this step, we have to identify the largest power of 2, which is less than or equal to the given number, 625. Since 2⁹ is the number we are looking for, write 1 in the 2⁹ place. Now, the value of 2⁹, which is 512, is subtracted from 625. 625 - 512 = 113.

Step 3 - Identify the next largest power of 2: In this step, we need to find the largest power of 2 that fits into the result of the previous step, 113. So, the next largest power of 2 is 2⁶, which is 64. Now, we have to write 1 in the 2⁶ place. And then subtract 64 from 113. 113 - 64 = 49.

Step 4 - Repeat the process: Continue identifying the largest powers of 2 that fit into the remainder until you reach 0. 2⁵ = 32, 49 - 32 = 17. 2⁴ = 16, 17 - 16 = 1. 2⁰ = 1, 1 - 1 = 0.

Step 5 - Identify the unused place values: In step 2 through step 4, we wrote 1s in the 2⁹, 2⁶, 2⁵, 2⁴, and 2⁰ places. Now, we can just write 0s in the remaining places, which are 2⁸, 2⁷, 2³, 2², and 2¹. Now, by substituting the values, we get: 0 in the 2⁸ place 0 in the 2⁷ place 1 in the 2⁶ place 1 in the 2⁵ place 1 in the 2⁴ place 0 in the 2³ place 0 in the 2² place 0 in the 2¹ place 1 in the 2⁰ place

Step 6 - Write the values in reverse order: We now write the numbers upside down to represent 625 in binary. Therefore, 1001110001 is 625 in binary.

Grouping Method: In this method, we divide the number 625 by 2. Let us see the step-by-step conversion.

Step 1 - Divide the given number 625 by 2. 625 / 2 = 312. Here, 312 is the quotient and 1 is the remainder.

Step 2 - Divide the previous quotient (312) by 2. 312 / 2 = 156. Here, the quotient is 156 and the remainder is 0.

Step 3 - Repeat the previous step. 156 / 2 = 78. Now, the quotient is 78, and 0 is the remainder.

Step 4 - Repeat the previous step. 78 / 2 = 39. Here, the quotient is 39 and the remainder is 0.

Step 5 - Repeat the previous step. 39 / 2 = 19. Here, the quotient is 19 and the remainder is 1.

Step 6 - Repeat the previous step. 19 / 2 = 9. Here, the quotient is 9 and the remainder is 1.

Step 7 - Repeat the previous step. 9 / 2 = 4. Here, the quotient is 4 and the remainder is 1.

Step 8 - Repeat the previous step. 4 / 2 = 2. Here, the quotient is 2 and the remainder is 0.

Step 9 - Repeat the previous step. 2 / 2 = 1. Here, the quotient is 1 and the remainder is 0.

Step 10 - Repeat the previous step. 1 / 2 = 0. Here, the remainder is 1. And we stop the division here because the quotient is 0.

Step 11 - Write down the remainders from bottom to top. Therefore, 625 (decimal) = 1001110001 (binary).

There are certain rules to follow when converting any number to binary. Some of them are mentioned below:

Rule 1: Place Value Method

This is one of the most commonly used rules to convert any number to binary. The place value method is the same as the expansion method, where we need to find the largest power of 2. Let’s see a brief step-by-step explanation to understand the first rule. Find the largest power of 2 less than or equal to 625. Since the answer is 2⁹, write 1 next to this power of 2. Subtract the value (512) from 625. So, 625 - 512 = 113. Find the largest power of 2 less than or equal to 113. The answer is 2⁶. So, write 1 next to this power. Continue this process until you reach a remainder of 0. Final conversion will be 1001110001.

Rule 2: Division by 2 Method

The division by 2 method is the same as the grouping method. A brief step-by-step explanation is given below for better understanding. First, 625 is divided by 2 to get 312 as the quotient and 1 as the remainder. Now, 312 is divided by 2. Here, we will get 156 as the quotient and 0 as the remainder. Continue dividing the subsequent quotients by 2 until the quotient becomes 0. Now, we write the remainders upside down to get the binary equivalent of 625, 1001110001.

Rule 3: Representation Method

This rule also involves breaking of the number into powers of 2. Identify the powers of 2 and write it down in decreasing order i.e., 2⁹, 2⁸, 2⁷, 2⁶, 2⁵, 2⁴, 2³, 2², 2¹, and 2⁰. Find the largest power that fits into 625. Repeat the process and allocate 1s and 0s to the suitable powers of 2. Combine the digits (0 and 1) to get the binary result.

Rule 4: Limitation Rule

The limitation of the binary system is that only 0s and 1s can be used to represent numbers. The system doesn’t use any other digits other than 0 and 1. This is a base 2 number system, where the binary places represent powers of 2. So, every digit is either a 0 or a 1. To convert 625, we use 0s for 2⁸, 2⁷, 2³, 2², and 2¹ and 1s for 2⁹, 2⁶, 2⁵, 2⁴, and 2⁰.

Learning a few tips and tricks is a great way to solve any mathematical problems easily. Let us take a look at some tips and tricks for binary numbers up to 625.

Memorize to speed up conversions: We can memorize the binary forms for numbers through practice.

Recognize the patterns: There is a peculiar pattern when converting numbers from decimal to binary.

Even and odd rule: Whenever a number is even, its binary form will end in 0. For example, 4 is even and its binary form is 100. Here, the binary of 4 ends in 0. If the number is odd, then its binary equivalent will end in 1. For example, the binary of 5 (an odd number) is 101. As you can see, the last digit here is 1.

Cross-verify the answers: Once the conversion is done, we can cross-verify the answers by converting the number back to the decimal form. This will eliminate any unforeseen errors in conversion.

Practice by using tables: Writing the decimal numbers and their binary equivalents on a table will help us remember the conversions.

• Decimal: The base 10 number system which uses digits from 0 to 9.

• Binary: This number system uses only 0 and 1. It is also called the base 2 number system.

• Place value: Every digit has a value based on its position in a given number. For example, in 625 (base 10), 6 is in the hundreds place, 2 in the tens place, and 5 in the ones place.

• Octal: It is the number system with a base of 8. It uses digits from 0 to 7.

• Power of 2: Refers to numbers like 2⁰, 2¹, 2², etc., which are used to determine the place values in binary representation.