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In this video, we will look at how to perform multiplications, both without any tools, and with a line-up/an alignment as our tool.
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We will do a number of examples, mainly on how to use the alignment method.
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When multiplying, the numbers that are being multiplied are called factors, and the result is called the product.
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So if we multiply 3 by 5, both 3 and 5 are factors and the result 15 is our product.
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When performing a multiplication, it does not matter in what order the factors are placed.
This is because 5 · 3 is the exact same as 3 · 5
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We begin by looking at a method on how to multiply using only our heads as our tool.
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What we do is to divide the largest factor into hundreds,
tens, ones, etc., and then multiply all these with the smaller factor.
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Let's do an example on this where we multiply 224 · 6
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To make it easier we can start by writing the smallest factor first, in this case 6 · 224
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We now separate the largest factor and thereby get: 6 · 200 + 20 · 6 + 6 · 4
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So, we divide the larger number into hundreds, tens and ones, then we multiply by the smaller factor, and lastly we add all these up.
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If we calculate each part separately we will get 1200 + 120 + 24
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After adding the first two terms together we get 1320 + 24
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At last we add 24 and we are left with our final answer 1344
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This method works fine with easier multiplications, but when it comes to more complicated ones, it can be good to know how to use the line-up method, which we will look at now.
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First, we will look at how to multiply integers using the line-up.
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We will start by calculating 112 · 3
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We begin by placing the factor with the most amount of digits, above the other factor.
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The idea here is to multiply the number below, with first the ones, then the tens, the hundreds etc. in the number above.
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So, first we multiply 3 by 2 and get 6
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Then we multiply 3 by 1 and get 3, and finally we multiply 3 by 1
and get 3 again.
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Here, the answer is 336
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That was a fairly simple example, but in some multiplications, we have to use a "memory-number", and that can get a bit more complicated.
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We will do an example of this by calculating 432 · 8. Like last time, we begin by lining up the numbers vertically.
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The first step is to multiply 8 by 2, which is 16. We now split 16 into tens and ones, and move the 1, representing the tens, to the side. The 6, representing the ones, we place below the ones in our line-up.
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The 1, that we moved to the side, is our so-called memory-number. This number represents the tens, which we can use in our next step.
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Next step is to multiply 8 by 3 to get 24
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We now want to use our memory-number, so that we can cross it over. To do this we add the 1 to the 24, since they both represent tens. We thereby get 24 + 1 = 25.
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We now split up 25, in the same way we did with the 16. So, we separate it in tens and hundreds (since 25 represents the amount of tens i.e. 250) This time 2 is our memory-number, and 5 is placed below the tens.
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We move on and multiply 8 by 4, which is equal to 32
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Again we add our memory-number 2 to 34 = 36, and we separate 36 in the same way as before. Our memory-number is now 3
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We have 3 as our last memory-number, and this we place to the far left.
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Our result of this given example 432 · 8, is thereby 3456
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We will do a final example in this video. We are going to calculate 612·31
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Again, we place the number with the most amount of digits above the other.
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In this case, our number below, 31, contains two digits. Let's split 31 in two, and first multiply the 1 with the number above, and then we do the same with the 3
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After multiplying both 1 and 3 with all the digits in the number above, we add these two products together.
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Let me visualize these steps, to get more clear.
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First, multiply 1 by 2 = 2, then 1 by 1 = 1
and lastly 1 by 6 = 6
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We move along and multiply 3 by 2 = 6
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We place the products of these multiplication below 612, and under the 3 in 31
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Then we multiply 3 by 1 = 3, and the last 3 by 6 = 18
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In the next step, we will sum these up, or in other words add 612 to 1836.
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We get 2 + 0 = 2, 1 + 6 = 7, 6 + 3 = 9, 0 + 8 = 8 and lastly
0 + 1 = 1
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Our final answer for 612 · 31 is 18 972